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.

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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

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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. \(\color{blue}{2}\)

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