How to Solve Logarithmic Equations? (+FREE Worksheet!)
In this blog post, you will learn how to solve Logarithmic Equations using the properties of logarithms in a few easy steps.
Related Topics
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.
- Plug the answers back into the original equation and check if the solution works.
The Absolute Best Books to Ace Pre-Algebra to Algebra II
Logarithmic Equations – Example 1:
Find the value of the variables in each equation. \(\log_{4}{(20-x^2)}=2\)
Solution:
Use log rule: \(\log_{b}{x}=\log_{b}{y}\), then: \(x=y\)
\(2=\log_{4}{4^2},\log_{4}{(20-x^2)}=\log_{4}{4^2}=\log_{4}{16}\)
then: \(20-x^2=16→20-16=x^2→x^2=4→x=2\) or \(x=-2\)
Logarithmic Equations – Example 2:
Find the value of the variables in each equation. \(log(2x+2)=log(4x-6)\)
Solution:
When the logs have the same base: \(f(x)=g(x)\),then: \(ln(f(x))=ln(g(x))\),
\(log(2x+2)=log(4x-6)→2x+2=4x-6→2x+2-4x+6=0\)
\(2x+2-4x+6=0→-2x+8=0→-2x=-8→x=\frac{-8}{-2}=4\)
Logarithmic Equations – Example 3:
Find the value of the variables in each equation. \(\log_{2}{(25-x^2)}=2\)
Solution:
Use log rule: \(\log_{b}{x}=\log_{b}{y}\), then: \(x=y\)
\(2=\log_{2}{2^2},\log_{2}{(25-x^2)}=\log_{2}{2^2}=\log_{2}{4}\)
Then: \(25-x^2=4→25-4=x^2→x^2=21 →x=\sqrt{21} \) or \(-\sqrt{21}\)
Logarithmic Equations – Example 4:
Find the value of the variables in each equation. \(log(8x+3)=log(2x-6)\)
Solution:
When the logs have the same base: \(f(x)=g(x)\),then: \(ln(f(x))=ln(g(x))\),
\(log(8x+3)=log(2x-6)→8x+3=2x-6→8x+3-2x+6=0\)
\(6x+9=0→6x=-9→x=\frac{-9}{6}=-\frac{3}{2}\)
Logarithms of negative numbers are not defined. Therefore, there is no solution for this equation.
Exercises for Logarithmic Equations
The Best Math Book to Help You Ace the Math Test
Find the value of the variables in each equation.
- \(\color{blue}{log(x+5)=2}\)
- \(\color{blue}{log x-log 4=3}\)
- \(\color{blue}{log x+log 2=4}\)
- \(\color{blue}{log 10+log x=1}\)
- \(\color{blue}{log x+log 8=log 48}\)
- \(\color{blue}{-3\log_{3}{(x-2)}=-12}\)
- \(\color{blue}{log 6x=log (x+5)}\)
- \(\color{blue}{log (4k-5)=log (2k-1)}\)
Answers
- \(\color{blue}{95}\)
- \(\color{blue}{4000}\)
- \(\color{blue}{5000}\)
- \(\color{blue}{1}\)
- \(\color{blue}{6}\)
- \(\color{blue}{83}\)
- \(\color{blue}{1}\)
- \(\color{blue}{2}\)
The Greatest Books for Students to Ace the Algebra
Related to This Article
More math articles
- Top 10 Free Websites for GED Math Preparation
- 5th Grade MEA Math Worksheets: FREE & Printable
- How to Graphs of Rational Functions?
- Laptop Buying Guide: Essential Tips to Know Before You Buy
- CLEP College Math-Test Day
- Cultivating a Love for Math: Parent’s Guide to Inspiring Passion
- FREE 6th Grade PARCC Math Practice Test
- How to Prepare for the HSPT Math Test?
- How to Use Partial Products to Multiply Two-Digit Numbers By Two-digit Numbers
- 5th Grade K-PREP Math Worksheets: FREE & Printable
What people say about "How to Solve Logarithmic Equations? (+FREE Worksheet!) - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.