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.

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

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Step by step guide to solve 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 in the answers back into the original equation and check to see the solution works.

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

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. \(\color{blue}{log (4k-5)=log (2k-1)}\)
This image has an empty alt attribute; its file name is answer-3.png

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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