How to Solve Logarithmic Equations? (+FREE Worksheet!)
To solve a logarithmic equation, combine the logs into a single log with the log rules, rewrite it in exponential form (logb x = y means by = x), solve, and check every answer, because the log of zero or a negative number is undefined. For example, log₂(x) + log₂(x − 2) = 3 becomes x(x − 2) = 8, so x = 4; the other root, x = −2, is rejected.
How to solve logarithmic equations, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.
In this blog post, you will learn how to solve Logarithmic Equations using the properties of logarithms in a few easy steps.
Solve Logarithmic Equations: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Translate firstAsk: the base to what power gives the input?
- Use rules legallyProducts, quotients, and powers have rules; sums do not split.
- Protect the domainKeep the log input positive and track asymptotes when graphing.
Worked examples
Evaluate a log
- Ask 3 to what power equals 81.
- 3 to the fourth power is 81.
- The logarithm is that exponent.
Find a log domain
- The input is x – 5.
- Require x – 5 > 0.
- Solve the inequality.
Try one before moving on
Solve Logarithmic Equations: pop-up practice
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.
For education statistics and research
Type the problems from this lesson and check each step.
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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
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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}{-3log_{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})
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