If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

## Course: Algebra 2 > Unit 8

- Solving exponential equations using logarithms: base-10

## Solving exponential equations using logarithms

- Solve exponential equations using logarithms: base-10 and base-e
- Solving exponential equations using logarithms: base-2
- Solve exponential equations using logarithms: base-2 and other bases

## Solving exponential equations of the form a ⋅ b x = d

Check your understanding.

- (Choice A) x = log 2 ( 39.3 ) A x = log 2 ( 39.3 )
- (Choice B) x = log 6 ( 118 ) B x = log 6 ( 118 )
- (Choice C) x = log 12 ( 236 ) C x = log 12 ( 236 )
- (Choice D) x = log 118 ( 6 ) D x = log 118 ( 6 )
- Your answer should be
- an integer, like 6
- a simplified proper fraction, like 3 / 5
- a simplified improper fraction, like 7 / 4
- a mixed number, like 1 3 / 4
- an exact decimal, like 0.75
- a multiple of pi, like 12 pi or 2 / 3 pi

## Solving exponential equations of the form a ⋅ b c x = d

- (Choice A) t = log ( 43.5 ) A t = log ( 43.5 )
- (Choice B) t = log 30 ( 130.5 ) B t = log 30 ( 130.5 )
- (Choice C) t = log ( 174 ) 4 C t = log ( 174 ) 4
- (Choice D) t = log 30 ( 522 ) 4 D t = log 30 ( 522 ) 4

## Challenge problem

- (Choice A) 2 A 2
- (Choice B) 3 B 3
- (Choice C) 4 C 4
- (Choice D) log 2 ( 3 ) D log 2 ( 3 )
- (Choice E) log 3 ( 2 ) E log 3 ( 2 )
- (Choice F) log ( 3 ) F log ( 3 )

## Want to join the conversation?

- Upvote Button navigates to signup page
- Downvote Button navigates to signup page
- Flag Button navigates to signup page

## HIGH SCHOOL

- ACT Tutoring
- SAT Tutoring
- PSAT Tutoring
- ASPIRE Tutoring
- SHSAT Tutoring
- STAAR Tutoring

## GRADUATE SCHOOL

- MCAT Tutoring
- GRE Tutoring
- LSAT Tutoring
- GMAT Tutoring
- AIMS Tutoring
- HSPT Tutoring
- ISEE Tutoring
- ISAT Tutoring
- SSAT Tutoring

## Search 50+ Tests

Loading Page

## math tutoring

- Elementary Math
- Pre-Calculus
- Trigonometry

## science tutoring

Foreign languages.

- Mandarin Chinese

## elementary tutoring

- Computer Science

## Search 350+ Subjects

- Video Overview
- Tutor Selection Process
- Online Tutoring
- Mobile Tutoring
- Instant Tutoring
- How We Operate
- Our Guarantee
- Impact of Tutoring
- Reviews & Testimonials
- Media Coverage
- About Varsity Tutors

## Precalculus : Solve Logarithmic Equations

Study concepts, example questions & explanations for precalculus, all precalculus resources, example questions, example question #1 : solve logarithmic equations.

Evaluate a logarithm.

The derifintion of logarithm is:

In this problem,

## Example Question #2 : Solve Logarithmic Equations

None of the other choices

Using the rules of logarithms,

So exponentiate both sides with a base 10:

The exponent and the logarithm cancel out, leaving:

This answer does not match any of the answer choices, therefore the answer is 'None of the other choices'.

## Example Question #1161 : Pre Calculus

Solve the following logarithmic equation:

In order to solve this equation, we must apply several properties of logarithms. First we notice the term on the left side of the equation, which we can rewrite using the following property:

Where a is the coefficient of the logarithm and b is some arbitrary base. Next we look at the right side of the equation, which we can rewrite using the following property for the addition of logarithms:

Using both of these properties, we can rewrite the logarithmic equation as follows:

We have the same value for the base of the logarithm on each side, so the equation then simplifies to the following:

## Example Question #1162 : Pre Calculus

None of the other answers.

We solve the equation as follows:

Exponentiate both sides.

Apply the power rule on the right hand side.

## Example Question #1163 : Pre Calculus

First, simplify the logarithmic expressions on the left side of the equation:

Now we have:

The left can be consolidated into one log expression using the subtraction rule:

We now have log on both sides, so we can be confident that whatever is inside these functions is equal:

## Example Question #1164 : Pre Calculus

First bring the inside exponent in front of the natural log.

Next simplify the first term and bring all the terms on one side of the equation.

Next, let set

## Example Question #1165 : Pre Calculus

Exponentiate each side to cancel the natural log:

Square both sides:

## Example Question #1166 : Pre Calculus

The base of a logarithm is 10 by default:

## Example Question #1167 : Pre Calculus

First, condense the left side into one logarithm:

## Example Question #1168 : Pre Calculus

no solution

First, consolidate the left side into one logarithm:

## Report an issue with this question

If you've found an issue with this question, please let us know. With the help of the community we can continue to improve our educational resources.

## DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC 101 S. Hanley Rd, Suite 300 St. Louis, MO 63105

Or fill out the form below:

## Contact Information

Complaint details.

## Find the Best Tutors

## How To Solve Logarithmic Equations

Step By Step Video and practice Problems

more interesting facts

## How To Solve Logarithmic Equations Video

## What is the general strategy for solving log equations?

Answer: As the video above points out, there are two main types of logarithmic equations. Before you to decide how to solve an equation, you must determine whether the equation

- A) has a logarithm on one side and a number on the other
- B) whether it has logarithms on both sides

## Example 1 Logarithm on one side and a number on the other

$$ log_4 x + log_4 8 = 3 $$

Step 1 Rewrite log side as single logarithm

$$ log_4 8x = 3 $$

Step 2 Rewrite as exponential equation

$$ 4^ 3 = 8x $$

Step 3 Solve the exponential equation

64 = 8x 8 = x

## Example 2 Logarithm on both sides

Step 1 use the rules of logarithms to rewrite the left side and the right side of the equation to a single logarithm

Step 2 "cancel" the log

Step 3 solve the expression

Let's look at a specific ex $$ log_5 x + log_2 3 = log_5 6 $$

Step 1 rewrite both sides as single logs

$$ log_5 x + log_5 2 = log_5 6 \\ log_5 2x = log_5 6 $$

Step 2 "cancel" logs

$$ \color{Red}{ \cancel {log_5}} 2x = \color{Red}{ \cancel {log_5}} 6 \\ 2x = 6 $$

Step 2 Solve expression

## Practice Problems

Solve the following equation: $ log_3 5 + log_3 x = log_3 15 $

Follow the steps for solving logarithmic equations with logs on both sides

rewrite both sides as single logs

$$ log_3 5x = log_3 15 $$

"cancel" logs

$$ \color{Red}{ \cancel{log_3}} 5x = \color{Red}{ \cancel{log_3}} 15 \\ 5x=15 $$

Solve expression

Solve the equation below: $ log_3 9 + log_3 x = 4 $

Follow the steps for solving logarithmic equations with a log on one side

Rewrite log side as single logarithm

$$ log_3 9x = 4$$

Rewrite as exponential equation

$$ 3^4 = 9x $$

Solve exponential equation

81 = 9 x 9 = x

Solve the following equation: $ 2log_3 5 + log_3 x = 3log_3 $

Follow the steps on how to solve equations with logs on both sides

rewrite both sides as single logs<

$ log_3 5^2 + log_3 x = log_3 5^3 \\ log_3 25 +log_3 x = log_3 125 \\ log_3 25x = log_3 125 $

$ \color{Red}{ \cancel{log_3}} 25x = \color{Red}{ \cancel{log_3}} 125 \\ 25x = 125 $

Solve the equation below: $ 2 log_2 4 + log_2 x = 5 $

$ 2 log_2 4 + log_2 x = 5 \\ log_2 4^2 = log_2 x = 5 \\ log_2 16 + log_2 x = 5 \\ log_2 16x = 5 $

32 =16x 2 = x

Solve the following equation: $ 2 log_3 5 + log_3 x = 3 log_3 5 $

$ log_3 5^2 + log_3 x + log_3 5^3 \\ log_3 25x + log_3 125 $

log 3 25x = log 3 5 3

$ \color{Red}{ \cancel{log_3}} 25x + \color{Red}{ \cancel{log_3}} 125 \\ 25x=125 $

$ \frac{25x}{25} = \frac{125}{25} \\ $

Solve the following equation: $2 log_3 7 - log_3 2x = log_3 98$

$ 2 log_3 7 - log_3 2x = log_3 98 \\ log_3 7^2 - log_3 2x = log_3 98 \\ log_3 49 - log_3 2x = log_3 98 \\ log_3 \frac{49}{2x} = log_3 98 $

$ \color{Red}{ \cancel{log_3}} \frac{49}{2x} = \color{Red}{ \cancel{log_3}} 98 \\ \frac{49}{2x} = 98 $

$ 49 = 196x \\ \frac{49}{196} = x \\ x = 49 $

Solve the following equation: $ 2 log_11 5 + log_11 x + log_11 2 = log_11 150 $

You know the deal. Just follow the steps for solving logarithmic equations with logs on both sides

rewrite as single logs

$ 2 log_11 5 + log_11 x + log_11 2 = log_11 150 \\ log_11 5^2 + log_11 2x = log_11 150 \\ log_11 25 + log_11 2x = log_11 150 \\ log_11 50x= log_11 150 $

2log 11 5 + log 11 x+ log 11 2 = log 11 150

$ \color{Red}{ \cancel{log_1}} 50x = \color{Red}{ \cancel{log_11}} 150 \\ 50x = 150 $

## Ultimate Math Solver (Free)

Free Algebra Solver ... type anything in there!

## Popular pages @ mathwarehouse.com and around the web

- Testimonial
- Web Stories

## Learning Home

## Not Now! Will rate later

- Logarithm: Solved Examples

- Logarithms: Concepts & Theory
- Logarithm: Practice Problems

## Most Popular Articles - PS

## Time and Work Concepts

## Time and Work Formula and Solved Problems

## Time and Work Problems (Easy)

## Time and Work Problems (Difficult)

## Problems on Ages Practice Problems : Level 02

## Chain Rule : Theory & Concepts

## Chain Rule Solved Examples

## Chain Rule Practice Problems: Level 01

## Chain Rule Practice Problems : Level 02

## Problems on Numbers System : Level 02

Download our app.

- Learn on-the-go
- Unlimited Prep Resources
- Better Learning Experience
- Personalized Guidance

Get More Out of Your Exam Preparation - Try Our App!

## IMAGES

## VIDEO

## COMMENTS

Here is a set of practice problems to accompany the Solving Logarithm Equations section of the Exponential and Logarithm Functions chapter of the notes for Paul Dawkins Algebra course at Lamar University.

8. Prove the following statements. (1) logp b x = 2log x (2) log p1 b p x = log x (3) log 4 x2 = log p x 9. Given that log2 = x, log3 = y and log7 = z, express the following expressions

900 possible mastery points Mastered Proficient Familiar Attempted Not started Quiz Unit test About this unit Logarithms are the inverses of exponents. They allow us to solve challenging exponential equations, and they are a good excuse to dive deeper into the relationship between a function and its inverse. Introduction to logarithms Learn

EXAMPLE 1 What is the result of \log_ {5} (x+1)+\log_ {5} (3)=\log_ {5} (15) log5(x +1) + log5(3) = log5(15)? Solution EXAMPLE 2 Solve the equation \log_ {4} (2x+2)+\log_ {4} (2)=\log_ {4} (x+1)+\log_ {4} (3) log4(2x+ 2) + log4(2) = log4(x + 1)+ log4(3) Solution EXAMPLE 3

Do 4 problems. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

Exercise 4.6e. 5. ★ For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation. 121. log( 1 100) = − 2. 122. log324(18) = 1 2. ★ For the following exercises, use the definition of a logarithm to solve the equation. 123. 5log7n = 10.

Since x = 7 x = 7 checks, we have a solution at \color {blue}x = 7 x = 7. Example 2: Solve the logarithmic equation. Start by condensing the log expressions on the left into a single logarithm using the Product Rule. We want to have a single log expression on each side of the equation.

Problem 1 Solve the equation \displaystyle \log_2 (x+2)=3 log2(x+2) = 3 Problem 2 Solve the equation \displaystyle \log_9 (3^x)=15 log9(3x) = 15 Problem 3 Solve the logarithmic equation: \displaystyle log_5x=3 log5x = 3 Problem 4 Solve the equation \displaystyle log_x36=2 logx36 = 2 Problem 5

The key to solving exponential equations lies in logarithms! Let's take a closer look by working through some examples. Solving exponential equations of the form a ⋅ b x = d Let's solve 5 ⋅ 2 x = 240 . To solve for x , we must first isolate the exponential part. To do this, divide both sides by 5 as shown below.

Section 1.9 : Exponential And Logarithm Equations. For problems 1 - 12 find all the solutions to the given equation. If there is no solution to the equation clearly explain why. 12−4e7+3x = 7 12 − 4 e 7 + 3 x = 7 Solution. 1 = 10−3ez2−2z 1 = 10 − 3 e z 2 − 2 z Solution. 2t−te6t−1 = 0 2 t − t e 6 t − 1 = 0 Solution.

Next we look at the right side of the equation, which we can rewrite using the following property for the addition of logarithms: Using both of these properties, we can rewrite the logarithmic equation as follows: We have the same value for the base of the logarithm on each side, so the equation then simplifies to the following:

exponential equation states 2 23 4 7x+ = ; it stands to reason, because of the equal sign, that 3 4 7x+ = . The problem would then be solved from this point. With logarithmic equations, the problem log 8 5 log 124 4(x− =) can be written as 8 5 12x − = . Always check for extraneous roots!!! Solve each of the following logarithmic equations.

Logarithmic Equations: Very Difficult Problems with Solutions Problem 1 Find the root of the equation \displaystyle 2+lg\sqrt {1+x}+3lg\sqrt {1-x}=lg\sqrt {1-x^2} 2+lg 1 +x +3lg 1−x = lg 1−x2 \displaystyle \frac {9} {100} 1009 \displaystyle \frac {99} {100} 10099 \displaystyle \frac {9} {10} 109 \displaystyle \frac {1} {9} 91

b logx +log(x−1) =log(3x+12) log x + log ( x − 1) = log ( 3 x + 12) Show Solution. c ln10−ln(7 −x) = lnx ln 10 − ln ( 7 − x) = ln x Show Solution. Now we need to take a look at the second kind of logarithmic equation that we'll be solving here. This equation will have all the terms but one be a logarithm and the one term that ...

Solving Logarithmic Equations (Word Problems) Example 1 INVESTMENT Mr. and Mrs. Mitchell are saving for their daughter's college education. They invest $10,000 in an account that pays 4.5% interest compounded continuously with the goal to have twice that amount in the account in ten years.

General method to solve this kind (logarithm on both sides), Step 1 use the rules of logarithms to rewrite the left side and the right side of the equation to a single logarithm. Step 2 "cancel" the log. Step 3 solve the expression. Let's look at a specific ex log5x + log23 = log56 l o g 5 x + l o g 2 3 = l o g 5 6.

Read the various tips to solve the logarithm questions in easy way. Solve the different practice problems to check your preparation level. ... Example 4: Solve for 'x:' the equation is 2log 2 x - log 2 (x - 2) = 3. A. 6. B. 4. C. 1. D. 2. Solution: We have 2log 2 x - log 2 (x - 2) = 3

15. hr. min. sec. SmartScore. out of 100. IXL's SmartScore is a dynamic measure of progress towards mastery, rather than a percentage grade. It tracks your skill level as you tackle progressively more difficult questions. Consistently answer questions correctly to reach excellence (90), or conquer the Challenge Zone to achieve mastery (100)!

Prev. Section Notes Practice Problems Assignment Problems Next Section Section 6.2 : Logarithm Functions For problems 1 - 3 write the expression in logarithmic form. 75 =16807 7 5 = 16807 Solution 163 4 = 8 16 3 4 = 8 Solution (1 3)−2 = 9 ( 1 3) − 2 = 9 Solution For problems 4 - 6 write the expression in exponential form.

4. Solve the following equation. log3(25−x2) = 2 log 3 ( 25 − x 2) = 2 Show All Steps Hide All Steps Start Solution

6.4 Solving Logarithm Equations; 6.5 Applications; 7. Systems of Equations. 7.1 Linear Systems with Two Variables; ... If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above. Section 6.2 : Logarithm Functions. For problems 1 - 5 write the expression in logarithmic form. ...

Practice Problems; Assignment Problems; Show/Hide; Show all Solutions/Steps/etc. Hide all Solutions/Steps/etc. Sections; Solving Exponential Equations ... Also, don't forget that the values with get when we are done solving logarithm equations don't always correspond to actual solutions to the equation so be careful! Start Solution. Recall ...