How to Solve Logarithmic Equations

How to Solve Logarithmic Equations

In this blog post, using the definition and rules of logarithms (exponentials and change-of-base formula), you will be taught how to solve logarithmic equations.

Related Topics

A step-by-step guide to solving Logarithmic Equations

  • Convert the logarithmic equation to an exponential equation when it’s possible. (If no base is indicated, the base of the logarithm is 10)
  • Condense logarithms if you have more than one log on one side of the equation.
  • Plugin the answers back into the original equation and check to see if the solution works.

Examples

Solving Logarithmic Equations – Example 1:

Find the value of \(x\) in this equation. \(log_{ 2}{(36-x^2)}=4\)

Solution:

Use log rule: \(log_{ b}{x}=log_{b}{ y}\), then: \(x=y\)
\(4=log_{2} {(2^4)} , log_{ 2}{(36-x^2 )}=log_{ 2}{(2^4 )}=log_{2}{ 16}\)
Then: \(36-x^2=16→36-16=x^2→x^2=20→x=\sqrt{20}=2\sqrt{5}\)

Solving Logarithmic Equations – Example 2:

Find the value of \(x\) in this equation. \(log⁡(5x+2)=log⁡(3x-1)\)

Solution:

When the logs have the same base: \(f(x)=g(x)\),then: \(ln(f(x))=ln(g(x))\), \(log⁡(5x+2)=log⁡(3x-1)→5x+2=3x-1→5x+2-3x+1=0→2x+3=0→2x=-3→x=-\frac{3}{2}\)
Verify Solution: \(log(5x+2)=log⁡(5(-\frac{3}{2})+2)=log⁡(-5.5) \)
Logarithms of negative numbers are not defined. Therefore, there is no solution for this equation.

Solving Logarithmic Equations – Example 3:

Find the value of the variables in this equation. \(log_{2}{(25-x^2)}=4\)

Solution:

Use the logarithmic definition: \(log_{a⁡}{(b}=c→a^c=b \)
\(log_{2}{(25-x^2 )}=4→2^4=(25-x^2 )→16=(25-x^2 ) \)
Simplify: \(16=(25-x^2 )→-x^2+25-16=0\)
Then: \(x^2=9→x=3\) or \(-3\) Both 3 and \(-3\) work in the original equation.

Solving Logarithmic Equations – Example 4:

Find the value of \(x\) in this equation. \(log⁡(3x+10)=log⁡(6x-8)\)

Solution:

When the logs have the same base: \(f(x)=g(x)\),then: \(ln(f(x))=ln(g(x))\), \(log⁡(3x+10)=log⁡(6x-8)→3x+10=6x-8→3x+10-10=6x-8-10→3x=6x-18→3x-6x=6x-18-6x→-3x=-18→x=\frac{-18}{-3}=6\)
Verify Solution: \(log(3x+10)=log⁡(3(6)+10)=log⁡(28) \)
Logarithms of negative numbers are not defined. Therefore, there is no solution for this equation.

Exercises for Solving Logarithmic Equations

Solve Logarithmic Equations.

  1. \(\color{blue}{log_{5}{3x}=0,x=}\)
  2. \(\color{blue}{log_{2}{6x}=2,x=}\)
  3. \(\color{blue}{(logx)+3=1,x=}\)
  4. \(\color{blue}{log3x=log(x+1),x=}\)
  5. \(\color{blue}{log2-logx=0,x=}\)
  6. \(\color{blue}{log(2x-1)=log(4x-2),x=}\)
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{\frac{1}{3}}\)
  2. \(\color{blue}{\frac{2}{3}}\)
  3. \(\color{blue}{\frac{1}{100}}\)
  4. \(\color{blue}{\frac{1}{2}}\)
  5. \(\color{blue}{2}\)
  6. \(\color{blue}{No \ solution \ for \ x∈R}\)

Related to "How to Solve Logarithmic Equations"

How to Solve Logarithmic Equations
How to Solve Logarithmic Equations

Leave a Reply

36% OFF

Download Instantly

X

How Does It Work?

Find Books

1. Find eBooks

Locate the eBook you wish to purchase by searching for the test or title.

add to cart

2. Add to Cart

Add the eBook to your cart.

checkout

3. Checkout

Complete the quick and easy checkout process.

download

4. Download

Immediately receive the download link and get the eBook in PDF format.

Why Buy eBook From Effortlessmath?

Save money

Save up to 70% compared to print

Instantly download

Instantly download and access your eBook

help environment

Help save the environment

Access

Lifetime access to your eBook

Test titles

Over 2,000 Test Prep titles available

Customers

Over 80,000 happy customers

Star

Over 10,000 reviews with an average rating of 4.5 out of 5

Support

24/7 support

Anywhere

Anytime, Anywhere Access

Find Your Test

Schools, tutoring centers, instructors, and parents can purchase Effortless Math eBooks individually or in bulk with a credit card or PayPal. Find out more…