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## 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 ‍

• (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 ) ‍
• 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 ) ‍

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## 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

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:

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## How To Solve Logarithmic Equations

Step By Step Video and practice Problems

more interesting facts

## 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

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

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• Logarithm: Solved Examples

• Logarithms: Concepts & Theory
• Logarithm: Practice Problems

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#### IMAGES

1. Solving Logarithmic Equations Worksheet

2. Logarithmic Equations Worksheet With Answers

3. Solving Logarithmic Equations Worksheet

4. Solving Logarithmic Equations

5. Solving Logarithmic Equations

6. Logarithmic Function Formula

#### VIDEO

1. Solving logarithmic equations

2. Solving Logarithmic Equations

3. Logarithmic equation

4. Solving Logarithmic Equations

5. 5.4 Solving Logarithmic Equations

6. Solving logarithmic equations

1. Algebra

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.

2. PDF Worksheet: Logarithmic Function

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

3. Logarithms

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

4. Logarithmic Equations

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

5. Evaluate logarithms (practice)

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.

6. 4.6e: Exercises

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.

7. Solving Logarithmic Equations

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.

8. Logarithmic Equations: Problems with Solutions

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

9. Solving exponential equations using logarithms

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.

10. Calculus I

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.

11. Solve Logarithmic Equations

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:

12. LOGARITHMS

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.

13. Logarithmic Equations: Very Difficult Problems with Solutions

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

14. Algebra

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 ...

15. PDF Solving Logarithmic Equations (Word Problems)

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.

16. How To Solve Logarithmic Equations. Video Tutorial and Practice Problems

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.

17. Logarithm Examples

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

18. IXL

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)!

19. Algebra

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.

20. Algebra

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

21. Algebra

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. ...

22. Algebra

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 ...