How to Solve Logarithmic Equations? (+FREE Worksheet!)

Related Topics

• How to Solve Natural Logarithms Problems
• How to Evaluate Logarithms
• Logarithms Properties

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.

Solving Logarithmic Equations For education statistics and research National Center for Education Statistics.

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

Solving Logarithmic Equations

The Best Math Book to Help You Ace the Math Test

Find the value of the variables in each equation.

1. \(\color{blue}{log⁡(x+5)=2}\)
2. \(\color{blue}{log x-log 4=3}\)
3. \(\color{blue}{log x+log 2=4}\)
4. \(\color{blue}{log 10+log x=1}\)
5. \(\color{blue}{log x+log 8=log 48}\)
6. \(\color{blue}{-3\log_{3}{(x-2)}=-12}\)
7. \(\color{blue}{log 6x=log (x+5)}\)
8. \(\color{blue}{log (4k-5)=log (2k-1)}\)

Answers

1. \(\color{blue}{95}\)
2. \(\color{blue}{4000}\)
3. \(\color{blue}{5000}\)
4. \(\color{blue}{1}\)
5. \(\color{blue}{6}\)
6. \(\color{blue}{83}\)
7. \(\color{blue}{1}\)
8. \(\color{blue}{2}\)

The Greatest Books for Students to Ace the Algebra