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

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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}{-3log_{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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