In this blog post, you will learn how to solve Logarithmic Equations using the properties of logarithms in a few easy steps.
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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: \(20-x^2=16→20-16=x^2→x^2=4→x=2\)
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: \(?(?)=?(?)\),?ℎ??: \(??(?(?))=??(?(?))
???2?+2=???4?−6→2?+2=4?−6→2?+2−4?+6=0\)
\(2x+2-4x+6=0→-2x+8=0→-2x=-8→x=\frac{-8}{-2}=4\)
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: \(25-x^2=4→25-16=x^2→x^2=9→x=3\)
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: \(?(?)=?(?)\),?ℎ??: \(??(?(?))=??(?(?))\)
\(log(8x+3)=log(2x-6)→8x+3=2x-6→8x+3-2x+6=0\)
\(6x+9=0→6x=-9→x=\frac{-9}{6}=-\frac{3}{2}\)
Exercises for Logarithmic Equations
Find the value of the variables in each equation.
- \(log(x+5)=2\)
- \(log x-log 4=3\)
- \(log x+log 2=4\)
- \(log 10+log x=1\)
- \(log x+log 8=log 48\)
- \(-3\log_{3}{(x-2)}=-12\)
- \(log 6x=log (x+5)\)
- \( log (4k-5)=log (2k-1)\)
Answers
- {-1/1,000}
- {4,000}
- {5,000}
- {1}
- {6}
- {83}
- {1}
- {2}
