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

Related Topics

• How to Solve Natural Logarithms
• How to Use Properties of Logarithms
• How to Evaluate Logarithms

A 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 to see if the solution works.

Solving Logarithmic Equations – Example 1:

Find the value of (x) in this equation. (log_{ 2}{(36-x^2)}=4)

Solution:

Use log rule: (log_{ b}{x}=log_{b}{ y}), then: (x=y)
(4=log_{2} {(2^4)} , log_{ 2}{(36-x^2 )}=log_{ 2}{(2^4 )}=log_{2}{ 16})
Then: (36-x^2=16→36-16=x^2→x^2=20→x=sqrt{20} ) or (-sqrt{20})

Solving Logarithmic Equations – Example 2:

Find the value of (x) in this equation. (log⁡(5x+2)=log⁡(3x-1))

Solution:

When the logs have the same base: (f(x)=g(x)),then: (ln(f(x))=ln(g(x))), (log⁡(5x+2)=log⁡(3x-1)→5x+2=3x-1→5x+2-3x+1=0→2x+3=0→2x=-3→x=-frac{3}{2})
Verify Solution: (log(5x+2)=log⁡(5(-frac{3}{2})+2)=log⁡(-5.5) )
Logarithms of negative numbers are not defined. Therefore, there is no solution for this equation.

Solving Logarithmic Equations – Example 3:

Find the value of the variables in this equation. (log_{2}{(25-x^2)}=4)

Solution:

Use the logarithmic definition: (log_{a⁡}{b}=c→a^c=b ),
(log_{2}{(25-x^2 )}=4→2^4=(25-x^2 )→16=(25-x^2 ) )
Simplify: (16=(25-x^2 )→-x^2+25-16=0)
Then: (x^2=9→x=3) or (-3), Both (3) and (-3) work in the original equation.

Solving Logarithmic Equations – Example 4:

Find the value of (x) in this equation. (log⁡(3x+10)=log⁡(6x-8))

Solution:

When the logs have the same base: (f(x)=g(x)),then: (ln(f(x))=ln(g(x))), (log⁡(3x+10)=log⁡(6x-8)→3x+10=6x-8→3x+10-10=6x-8-10→3x=6x-18→3x-6x=6x-18-6x→-3x=-18→x=frac{-18}{-3}=6)
Verify Solution: (log(3x+10)=log⁡(3(6)+10)=log⁡(28) )

Exercises for Solving Logarithmic Equations

Solve Logarithmic Equations.

1. (color{blue}{log_{5}{3x}=0,x=})
2. (color{blue}{log_{2}{6x}=2,x=})
3. (color{blue}{(logx)+3=1,x=})
4. (color{blue}{log3x=log(x+1),x=})
5. (color{blue}{log2-logx=0,x=})
6. (color{blue}{log(2x-1)=log(4x-2),x=})

1. (color{blue}{frac{1}{3}})
2. (color{blue}{frac{2}{3}})
3. (color{blue}{frac{1}{100}})
4. (color{blue}{frac{1}{2}})
5. (color{blue}{2})
6. (color{blue}{No solution for x∈R})