Solving Logarithmic Equations

Do you want to know how to solve Logarithmic Equations? you can do it in two easy steps.

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.

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: \(20-x^2=16→20-16=x^2→x^2=4→x=2\)

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\)

\(2x+2-4x+6=0→-2x+8=0→-2x=-8→x=\frac{-8}{-2}=4\)

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: \(25-x^2=4→25-16=x^2→x^2=9→x=3\)

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

Find the value of the variables in each equation.

1. \(log⁡(x+5)=2\)
2. \(log x-log 4=3\)
3. \(log x+log 2=4\)
4. \(log 10+log x=1\)
5. \(log x+log 8=log 48\)
6. \(-3\log_{3}{(x-2)}=-12\)
7. \(log 6x=log (x+5)\)
8. \( log (4k-5)=log (2k-1)\)

Answers

1. {-1/1,000}
2. {4,000}
3. {5,000}
4. {1}
5. {6}
6. {83}
7. {1}
8. {2}