Section 1.5: Exponential and Logarithmic Functions

Section 1.5:  Part 1

This content comes directly from West Texas A&M University’s College Algebra Tutorial 42: Exponential Functions, revised by Kim Seward.

Access this resource for free at https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut42_expfun.htm


Learning Objectives

In this section, you will:

  • Use the exponent and [latex]e[/latex] keys on your calculator.
  • Know the definition of an exponential function.
  • Evaluate an exponential function.
  • Graph exponential functions.

Let’s Get Started…

One thing we will be looking at in this section is using your exponent and [latex]e[/latex] keys on your calculator. So make sure that you have your calculator ready to go.  It will also help you with the graphing problems.  Let’s take a look at these exponential functions.

 

Exponential Function

The function[latex]f[/latex] defined by

[latex]f\left(x\right)={b}^{x}[/latex]

where [latex]b[/latex] > [latex]0[/latex], [latex]b\ne1[/latex], and the exponent [latex]x[/latex] is any real number, is called an exponential function.

 

Again, note that the variable [latex]x[/latex] is in the exponent, as opposed to the base, when we are dealing with an exponential function.

Also, note that in this definition, the base [latex]b[/latex] is restricted to being a positive number other than 1.

 

Exponent Key on Calculator

Before we get going with the actual exponential functions, I wanted to make sure that everyone knew how to use the exponent key on their calculator. Since there are a lot of different calculators, I will be go over the more common ones.

The two main exponent keys found on calculators are (you will only have one of these):

  1. Caret top key: ^
  1. Base raised to an exponent key: [latex]{x}^{y}[/latex] or [latex]{y}^{x}[/latex]

 

The caret top key is most frequently found on graphing calculators but can be found on other types of calculators.

 

How To

Use the caret top key to calculate 15 raised to the 5th power.

  1. Type in 15^5.
  2. Press enter or the equal key.
  3. If you got 759375, then you used the caret top key correctly.

 

The key that looks like [latex]{x}^{y}[/latex] or [latex]{y}^{x}[/latex] is most common in business and scientific calculators but can be found on other types of calculators.

 

How To

Use [latex]{x}^{y}[/latex] or [latex]{y}^{x}[/latex] to calculate 15 raised to the 5th power.

  1. Type in 15.
  2. Press the [latex]{x}^{y}[/latex] or [latex]{y}^{x}[/latex] key.
  3. Type in 5.
  4. Press enter or the equal key.
  5. If you got 759375, then you used the caret top key correctly.

 

Base [latex]e[/latex]

The Natural Exponential Function

The exponential function[latex]f[/latex] with base [latex]e[/latex],

[latex]f\left(x\right)={e}^{x}[/latex],

is called the natural exponential function.  [latex]e[/latex] is a special number with a value of approximately 2.718281828…

 

[latex]e[/latex] Key on a Calculator

I want to make sure that everyone knows how to use the [latex]e[/latex] key on their calculator.  Since there are a lot of different calculators, I will be go over the more common ones.

The two main exponent keys found on calculators are (you will only have one of these):

  1. [latex]e[/latex] key and the caret top key (2 keys): [latex]e[/latex] and then  ^
  1. [latex]e[/latex]raised to an exponent key (1 key): [latex]{e}^{x}[/latex]
The [latex]e[/latex] key with the caret top key is most frequently found on graphing calculators but can be found on other types of calculators.

 

How To

Use [latex]e[/latex] key with the caret top key to calculate [latex]e[/latex] raised to the 5th power.

  1. Press the [latex]e[/latex] key.
  2. Press the ^ key.
  3. Type 5.  (You should see [latex]{e}^{5}[/latex] on your screen.)
  4. Press enter or the equal key.
  5. If you got 148.41316…, you entered it in correctly.

 

The key that looks like [latex]{e}^{x}[/latex] is most common in business and scientific calculators, but can be found on other types of calculators.  Note that the difference between this key and the [latex]e[/latex] key above is that [latex]{e}^{x}[/latex] has a variable exponent showing on the key. The [latex]e[/latex] key above only has an [latex]e[/latex] with no exponent.

 

How To

Use the [latex]{e}^{x}[/latex] key to calculate [latex]e[/latex] raised to the 5th power.

  1. Type in 5.
  2. Press the [latex]{e}^{x}[/latex] key.
  3. Press enter or the equal key.
  4. If you got 148.41316…, you entered it in correctly.

 

Using the Calculator, in General

As mentioned above, there are a lot of different types of calculators out there.  I want to mention a few things about putting in formulas.

Graphing Calculators:

Most graphing calculators allow you to put in the whole formula before you press enter.  In fact, you are able to see it all.  If you are going to plug in the whole formula at one time, just make sure you are careful.  Pay special attention to putting in the parenthesis in the right place.

 

Business and Scientific Calculators:

On most business and scientific calculators, you will have to put the formula in part by part.  Work your way inside out of the parenthesis. DO NOT round until you are at the end.  As you go step by step, don’t erase what you have on your calculator screen.  Use it in the next step so you will have the full decimal number.

 

All Calculators:

DO NOT round until you get to the final answer. You will note on a lot of the examples that I put dots after my numbers that would keep going on an on if I had more space on my calculator.  Keep in mind that your calculator may have fewer or more spaces than my calculator does – so your calculator may have a slightly different answer than mine, due to rounding.  It should be very close though.

 


EXAMPLE 1

Performing exponential calculations with [latex]e[/latex]

Approximate the number [latex]{e}^{7.25}[/latex] using a calculator. Round to four decimal places.

 

Show/Hide Solution

 


EXAMPLE 2

Performing exponential calculations with [latex]e[/latex]

Approximate the number [latex]{e}^{-0.12}[/latex] using a calculator. Round to four decimal places.

 

Show/Hide Solution

 


Graphing Exponential Functions

The process of graphing any exponential function follows the same 3 basic steps.

 

Step 1: Find ordered pairs.

I have found that the best way to do this is to do it the same each time.  In other words, put in the same values for [latex]x[/latex] each time and then find the corresponding [latex]y-[/latex]value for the given function.

 

Step 2: Plot points.

This is done exactly the same way you plotted points when you graphed lines and parabolas.

 

Step 3: Draw curve.

The basic curve of an exponential function looks like the following:

 

curve 1ORcurve 2

 


EXAMPLE 3

Graphing an exponential function

Graph the function [latex]f\left(x\right)={4}^{x}[/latex].

 

Show/Hide Solution

 

Step 1: Find ordered pairs.

I have found that the best way to do this is to do it the same way each time.  Let’s substitute the values [latex]-2, -1, 0, 1, \text{and } 2[/latex] for [latex]x[/latex] to find their corresponding [latex]f\left(x\right)[/latex] values.
[latex]x[/latex] [latex]f\left(x\right)={4}^{x}[/latex]
[latex]-2[/latex] [latex]f\left(-2\right)={4}^{-2}=\frac{1}{{4}^{2}}=\frac{1}{16}=0.0625[/latex]
[latex]-1[/latex] [latex]f\left(-1\right)={4}^{-1}=\frac{1}{{4}^{1}}=\frac{1}{4}=0.25[/latex]
[latex]0[/latex] [latex]f\left(0\right)={4}^{0}=1[/latex]
[latex]1[/latex] [latex]f\left(1\right)={4}^{1}=4[/latex]
[latex]2[/latex] [latex]f\left(2\right)={4}^{2}=16[/latex]

This gives the ordered pairs: [latex]\left(-2,0.0625\right), \left(-1,0.25\right), \left(0,1\right), \left(1,4\right), \text{and} \left(2,16\right).[/latex]

 

Step 2: Plot points AND Step 3: Draw curve.

First, plot the 5 points above on a set of [latex]x-y[/latex] axes. Then, connect the points with a smooth, exponential curve.

example 3h

 


EXAMPLE 4

Graphing an exponential function

Graph the function [latex]f\left(x\right)={4}^{x-1}+3[/latex].

 

Show/Hide Solution

Step 1: Find ordered pairs.

Again, let’s substitute the values [latex]-2, -1, 0, 1, \text{and } 2[/latex] for [latex]x[/latex] to find their corresponding [latex]f\left(x\right)[/latex] values.
[latex]x[/latex] [latex]f\left(x\right)={4}^{x-1}+3[/latex]
[latex]-2[/latex] [latex]f\left(-2\right)={4}^{-2-1}+3=\frac{1}{{4}^{3}}+3=\frac{1}{64}+3=3.015625[/latex]
[latex]-1[/latex] [latex]f\left(-1\right)={4}^{-1-1}+3=\frac{1}{{4}^{2}}+3=\frac{1}{16}+3=3.0625[/latex]
[latex]0[/latex] [latex]f\left(0\right)={4}^{0-1}+3=\frac{1}{{4}^{1}}+3=\frac{1}{4}+3=3.25[/latex]
[latex]1[/latex] [latex]f\left(1\right)={4}^{1-1}+3={4}^{0}+3=1+3=4[/latex]
[latex]2[/latex] [latex]f\left(2\right)={4}^{2-1}+3={4}^{1}+3=4+3=7[/latex]

This gives the ordered pairs: [latex]\left(-2,3.015625\right), \left(-1,3.0625\right), \left(0,3.25\right), \left(1,4\right), \text{and} \left(2,7\right).[/latex]

 

Step 2: Plot points AND Step 3: Draw curve.

First, plot the 5 points above on a set of [latex]x-y[/latex] axes. Then, connect the points with a smooth, exponential curve.

 

example 4h

 


EXAMPLE 5

Graphing an exponential function

Graph the function [latex]f\left(x\right)=4\left(\frac{1}{4}\right)^{x}-3[/latex].

 

Show/Hide Solution

Step 1: Find ordered pairs.

Like the last 2 examples, let’s substitute the values [latex]-2, -1, 0, 1, \text{and } 2[/latex] for [latex]x[/latex] to find their corresponding [latex]f\left(x\right)[/latex] values.
[latex]x[/latex] [latex]f\left(x\right)=4\left(\frac{1}{4}\right)^{x}-3[/latex]
[latex]-2[/latex] [latex]f\left(-2\right)=4\left(\frac{1}{4}\right)^{-2}-3=4\left({4}\right)^{2}-3=4\left(16\right)-3=64-3=61[/latex]
[latex]-1[/latex] [latex]f\left(-1\right)=4\left(\frac{1}{4}\right)^{-1}-3=4\left({4}\right)^{1}-3=4\left(4\right)-3=16-3=13[/latex]
[latex]0[/latex] [latex]f\left(0\right)=4\left(\frac{1}{4}\right)^{0}-3=4\left(1\right)-3=4-3=1[/latex]
[latex]1[/latex] [latex]f\left(1\right)=4\left(\frac{1}{4}\right)^{1}-3=4\left(\frac{1}{4}\right)-3=1-3=-2[/latex]
[latex]2[/latex] [latex]f\left(2\right)=4\left(\frac{1}{4}\right)^{2}-3=4\left(\frac{1}{16}\right)-3=\frac{4}{16}-3=-2.75[/latex]

This gives the ordered pairs: [latex]\left(-2,61\right), \left(-1,13\right), \left(0,1\right), \left(1,-2\right), \text{and} \left(2,-2.75\right).[/latex]

 

Step 2: Plot points AND Step 3: Draw curve.

First, plot the 5 points above on a set of [latex]x-y[/latex] axes. Then, connect the points with a smooth, exponential curve.example 5h

 


Section 1.5:  Part 2

This content comes directly from West Texas A&M University’s College Algebra Tutorial 43: Logarithmic Functions, revised by Kim Seward.

Access this resource for free at https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut43_logfun.htm


Learning Objectives

In this section, you will:

  • Know the definition of a logarithmic function.
  • Write a log function as an exponential function and vice versa.
  • Graph a log function.
  • Evaluate a log.
  • Find the domain of a log function.

Let’s Get Started…

In this tutorial, we will be looking at logarithmic functions. If you understand that A LOG IS ANOTHER WAY TO WRITE AN EXPONENT, it will help you tremendously when you work through the various types of log problems.  One thing that I will guide you through on this page is the definition of logs.  This is an important concept to have down.  If you don’t have it down, it makes it hard to work through log related problems.  I will also take you through graphing, evaluating, and finding the domain of logs.

 

Logarithmic Function

The logarithmic function with base [latex]b[/latex], where [latex]b\,[/latex]>[latex]\,0[/latex] and [latex]b\ne1[/latex], is defined by

[latex]y=\log_{b}\left(x\right)[/latex]

if and only if

[latex]{b}^{y}=x.[/latex]

 

IN OTHER WORDS – AND I CAN NOT STRESS THIS ENOUGH – A LOG IS ANOTHER WAY TO WRITE AN EXPONENT. 

This definition can work in both directions.  In some cases you will have an equation written in log form and need to convert it to exponential form and vice versa.

So, when you are converting from log form to exponential form, [latex]b[/latex] is your base, [latex]y[/latex] IS YOUR EXPONENT, and [latex]x[/latex] is what your exponential expression is set equal to.

Note that your domain is all positive real numbers and range is all real numbers.

 


EXAMPLE 1

Expressing logarithms as exponents

Express the logarithmic equation [latex]3=\log_{5}\left(125\right)[/latex] exponentially.

 

Show/Hide Solution

 


Hopefully, when you see it written in exponential form, you can tell that it is a true statement.  In other words, when we cube 5 we do get 125.  If you had written this as 5 raised to the 125th power, hopefully you would have realized that was not correct because it would not equal 3.

EXAMPLE 2

Expressing logarithms as exponents

Express the logarithmic equation [latex]\log_{7}\left(49\right)=y[/latex] exponentially.

 

Show/Hide Solution

 


EXAMPLE 3

Expressing exponentials as logarithms

Express the exponential equation [latex]{6}^{-2}=\frac{1}{36}[/latex] in logarithmic form.

 

Show/Hide Solution

 


EXAMPLE 4

Expressing exponentials as logarithms

Express the exponential equation [latex]\sqrt{81}=x[/latex] in logarithmic form.

 

Show/Hide Solution

Rewriting the original problem using exponents, we get:

 

[latex]\sqrt{81}={81}^{\frac{1}{2}}.[/latex]

 

 


Evaluating Logarithms

The process of evaluating logs follows the same 3 basic steps.

How To

Evaluate a logarithm.

  1. Set the log equal to [latex]x[/latex].
  2. Use the definition of logs to write the equation in exponential form.
  3. Find [latex]x[/latex].

 

EXAMPLE 5

Evaluating logarithms

Evaluate the expression [latex]\log_{6}\left(64\right)[/latex] without using a calculator.

 

Show/Hide Solution

Step 3: Find [latex]x[/latex]

Since 3 is the exponent of [latex]4[/latex] we need to get 64, we know that:

[latex]x=3.[/latex]

 


EXAMPLE 6

Evaluating logarithms

Evaluate the expression [latex]\log_{9}\left(1\right)[/latex] without using a calculator.

 

Show/Hide Solution

Step 3: Find [latex]x[/latex]

Since 0 is the exponent of [latex]9[/latex] we need to get 1, we know that:

[latex]x=0.[/latex]

 


EXAMPLE 7

Evaluating logarithms

Evaluate the expression [latex]\log_{7}\left(7\right)[/latex] without using a calculator.

 

Show/Hide Solution

Step 3: Find [latex]x[/latex]

Since 1 is the exponent of [latex]7[/latex] we need to get 7, we know that:

[latex]x=1.[/latex]

 


EXAMPLE 8

Evaluating logarithms

Evaluate the expression [latex]\log_{5}\left(\sqrt{5}\right)[/latex] without using a calculator.

 

Show/Hide Solution

Step 3: Find [latex]x[/latex]

Since [latex]\frac{1}{2}[/latex] is the exponent of [latex]5[/latex] we need to get [latex]\sqrt{5}[/latex], we know that:

[latex]x=\frac{1}{2}.[/latex]

 


Graphing Logarithmic Functions

The process of graphing any logarithmic function follows the same 4 basic steps.

 

Step 1: Use the definition of a logarithmic function to write it in exponential form.

You have to be careful that you note that the [latex]log[/latex] key on your calculator is only for base 10 and your [latex]ln[/latex] key is only for base [latex]e.[/latex] So if you have any other base, you would not be able to use your calculator. But, if you write a logarithm in exponential form, you can enter in any base in your calculator – that is why we do step 1.

 

Step 2: Plug in values for [latex]y[/latex] (NOT [latex]x[/latex]) to find some ordered pairs.

Note that this is what we call an inverse function of the exponential function. They are inverses because the [latex]x-[/latex] and [latex]y-[/latex]values are switched.  In the exponential functions, the [latex]x-[/latex]value was the exponent, but in the log functions, the [latex]y-[/latex]value is the exponent. The [latex]y-[/latex]value is what the exponential function is set equal to, but in the log functions it ends up being set equal to [latex]x.[/latex] So that is why in step 2, we will be plugging in for [latex]y[/latex] instead of [latex]x[/latex].

 

Step 3: Plot points.

This is done exactly the same way you plot points for any graph.

 

Step 4: Draw curve.

The basic curve of a logarithmic function looks like the following:

 

curve 2ORcurve 1

 


EXAMPLE 9

Graphing a logarithmic function

Graph the function [latex]y=f\left(x\right)=\log_{3}\left(x\right).[/latex]

 

Show/Hide Solution

Step 1: Use the definition of a logarithmic function to write it in exponential form.

Looks like the base is 3, the exponent is [latex]y[/latex], and the exponential form will be set = to [latex]x[/latex]:

[latex]{3}^{y}=x[/latex]

 

Step 2: Plug in values for [latex]y[/latex] (NOT [latex]x[/latex]) to find some ordered pairs.

I have found that the best way to do this is to do it the same way each time.  Let’s substitute the values [latex]-2, -1, 0, 1, \text{and } 2[/latex] for [latex]y[/latex] to find their corresponding [latex]{3}^{y}=x[/latex] values.
[latex]{3}^{y}=x[/latex] [latex]y[/latex]
[latex]{3}^{-2}=\frac{1}{{3}^{2}}=\frac{1}{9}[/latex] [latex]-2[/latex]
[latex]{3}^{-1}=\frac{1}{{3}^{1}}=\frac{1}{3}[/latex] [latex]-1[/latex]
[latex]{3}^{0}=1[/latex] [latex]0[/latex]
[latex]{3}^{1}=3[/latex] [latex]1[/latex]
[latex]{3}^{2}=9[/latex] [latex]2[/latex]

This gives the ordered pairs: [latex]\left(\frac{1}{9},-2\right), \left(\frac{1}{3},-1\right), \left(1,0\right), \left(3,1\right), \text{and} \left(9,2\right).[/latex]

 

Step 3: Plot points AND Step 4: Draw curve.

First, plot the 5 points above on a set of [latex]x-y[/latex] axes. Then, connect the points with a smooth, exponential curve.

example 9g

 


EXAMPLE 10

Graphing a logarithmic function

Graph the function [latex]y=f\left(x\right)=\log_{3}\left(x+1\right).[/latex]

 

Show/Hide Solution

Step 1: Use the definition of a logarithmic function to write it in exponential form.

Looks like the base is 3, the exponent is [latex]y[/latex], and the exponential form will be set = to [latex]x+1[/latex]:

[latex]{3}^{y}=x+1[/latex]

or

[latex]{3}^{y}-1=x[/latex]

 

Step 2: Plug in values for [latex]y[/latex] (NOT [latex]x[/latex]) to find some ordered pairs.

I have found that the best way to do this is to do it the same way each time.  Let’s substitute the values [latex]-2, -1, 0, 1, \text{and } 2[/latex] for [latex]y[/latex] to find their corresponding [latex]{3}^{y}-1=x[/latex] values.
[latex]{3}^{y}-1=x[/latex] [latex]y[/latex]
[latex]{3}^{-2}-1=\frac{1}{{3}^{2}}-1=\frac{1}{9}-1=-\frac{8}{9}[/latex] [latex]-2[/latex]
[latex]{3}^{-1}-1=\frac{1}{{3}^{1}}-1=\frac{1}{3}-1=-\frac{2}{3}[/latex] [latex]-1[/latex]
[latex]{3}^{0}-1=1-1=0[/latex] [latex]0[/latex]
[latex]{3}^{1}-1=3-1=2[/latex] [latex]1[/latex]
[latex]{3}^{2}-1=9-1=8[/latex] [latex]2[/latex]

This gives the ordered pairs: [latex]\left(-\frac{8}{9},-2\right), \left(-\frac{2}{3},-1\right), \left(0,0\right), \left(2,1\right), \text{and} \left(8,2\right).[/latex]

 

Step 3: Plot points AND Step 4: Draw curve.

First, plot the 5 points above on a set of [latex]x-y[/latex] axes. Then, connect the points with a smooth, exponential curve.

example 10h

 


EXAMPLE 11

Graphing a logarithmic function

Graph the function [latex]y=f\left(x\right)=-\log_{3}\left(x\right).[/latex]

 

Show/Hide Solution

Step 1: Use the definition of a logarithmic function to write it in exponential form.

First, let’s rewrite the function to make the side of the equation with the log positive:

[latex]y=-\log_{3}\left(x\right)\,\,\Rightarrow\,\,-y=\log_{3}\left(x\right).[/latex]

Now, it looks like the base is 3, the exponent is [latex]-y[/latex], and the exponential form will be set = to [latex]x[/latex]:

[latex]{3}^{-y}=x[/latex]

 

Step 2: Plug in values for [latex]y[/latex] (NOT [latex]x[/latex]) to find some ordered pairs.

I have found that the best way to do this is to do it the same way each time.  Let’s substitute the values [latex]-2, -1, 0, 1, \text{and } 2[/latex] for [latex]y[/latex] to find their corresponding [latex]{3}^{-y}=x[/latex] values.
[latex]{3}^{-y}=x[/latex] [latex]y[/latex]
[latex]{3}^{-\left(-2\right)}={3}^{2}=9[/latex] [latex]-2[/latex]
[latex]{3}^{-\left(-1\right)}={3}^{1}=3[/latex] [latex]-1[/latex]
[latex]{3}^{-0}={3}^{0}=1[/latex] [latex]0[/latex]
[latex]{3}^{-1}=\frac{1}{{3}^{1}}=\frac{1}{3}[/latex] [latex]1[/latex]
[latex]{3}^{-2}=\frac{1}{{3}^{2}}=\frac{1}{9}[/latex] [latex]2[/latex]

This gives the ordered pairs: [latex]\left(9,-2\right), \left(3,-1\right), \left(1,0\right), \left(\frac{1}{3},1\right), \text{and} \left(\frac{1}{9},2\right).[/latex]

 

Step 3: Plot points AND Step 4: Draw curve.

First, plot the 5 points above on a set of [latex]x-y[/latex] axes. Then, connect the points with a smooth, exponential curve.

example 11i

 


EXAMPLE 12

Finding the domain of logarithmic functions

Find the domain of the logarithmic function [latex]f\left(x\right)=\log_{2}\left(5-x\right).[/latex]

 

Show/Hide Solution

 


EXAMPLE 13

Finding the domain of logarithmic functions

Find the domain of the logarithmic function [latex]f\left(x\right)=\log_{2}{\left(2+x\right)}^{2}.[/latex]

 

Show/Hide Solution

 


Inverse Properties of Logarithms

There are 2 inverse properties of logs that can be powerful simplifying tools.

 

Inverse Property I

[latex]\log_{b}\left({b}^{r}\right)=r[/latex]

where[latex]\,b\,[/latex] > [latex]0[/latex] and[latex]\,b\ne1.[/latex]

 

Basically, what we are saying here is, whenever the base of your log matches with the base of the inside of your log, then the log will equal the exponent of the inside base – but only if the bases match!!!

Boy, the definition of logs sure does come in handy to explain these properties.  Applying that definition you would have [latex]b[/latex] raised to the [latex]r[/latex] power, which equals [latex]b[/latex] raised to the [latex]r[/latex] power.

Here is a quick illustration of how this property works:

[latex]\log_{5}\left({5}^{\frac{1}{2}}\right)=?[/latex]

[latex]\log_{5}\left({5}^{\frac{1}{2}}\right)=\frac{1}{2}.[/latex]

 

 

Inverse Property II

[latex]{b}^{\log_{b}\left(m\right)}=m[/latex]

where[latex]\,b\,[/latex] > [latex]0[/latex] and[latex]\,b\ne1.[/latex]

 

Basically, what we are saying here is, whenever you have a base raised to a log with the SAME base, then it simplifies to be whatever is inside the log.

This one is a little bit more involved and weird looking huh? Going back to our favorite saying – a log is another way to write exponents – what we have here is the log is the exponent we need to raise [latex]b[/latex] to get [latex]m[/latex]. Well, if we turn around and raise our first base [latex]b[/latex] to that exponent, it stands to reason that we would get [latex]b[/latex].

Here is a quick illustration of how this property works:

[latex]{2}^{\log_{2}\left(3\right)}=?[/latex]

[latex]{2}^{\log_{2}\left(3\right)}=3.[/latex]

 


Most Frequently Used Logarithms

There are 2 bases that are the most frequently used for logarithms.

Common Log

[latex]\log\left(x\right)\,\,\Rightarrow\,\,\log_{10}\left(x\right)[/latex]

 

 

In other words, if no base is written for the log, it is understood to be base 10, which is called the common log.

When using common log (base 10), use the form [latex]\log\left(x\right)[/latex] to write it.

 

Natural Log

[latex]\ln\left(x\right)\,\,\Rightarrow\,\,\log_{e}\left(x\right)[/latex]

 

 

In other words, if the log is written with ln, instead of log in front of the [latex]x[/latex], then it is understood to be a log of base [latex]e[/latex],which is called the natural log.

When using the natural log (base [latex]e[/latex]), use the form [latex]\ln\left(x\right)[/latex] to write it.

 


EXAMPLE 14

Evaluating logarithms

Evaluate [latex]\log\left(0.001\right)[/latex] without using a calculator.

 

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EXAMPLE 15

Evaluating logarithms

Evaluate [latex]{e}^{\ln\left(50\right)}[/latex] without using a calculator.

 

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[latex]{e}^{\ln\left(50\right)}=50[/latex]

 


EXAMPLE 16

Evaluating logarithms

Evaluate [latex]{10}^{\log\left({5x}^{2}\right)}[/latex] without using a calculator.

 

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[latex]{10}^{\log\left({5x}^{2}\right)}={5x}^{2}[/latex]

 


EXAMPLE 17

Evaluating logarithms

Evaluate [latex]\ln\left(\frac{1}{{e}^{x}}\right)[/latex] without using a calculator.

 

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Baylor University's Co-requisite Supplement for Calculus I Copyright © 2023 by Amy Graham Goodman is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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