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
- Math Skills You Need for the GED Math Test
- How to Study Math Effectively?
- Top 10 5th Grade Georgia Milestones Assessment System Math Practice Questions
- Back to School: The Best Laptops for Students
- The Ultimate 7th Grade PARCC Math Course (+FREE Worksheets)
- Is a Calculator Allowed on the CBEST Test?
- 5th Grade IAR Math Practice Test Questions
- The Fundamental Theorem of Algebra
- How Hard Is the ALEKS Math Test?
- Getting a Math Degree: Hacks to Make Your Life Easier
What people say about "How to Solve Logarithmic Equations? (+FREE Worksheet!) - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.