In this blog post, using the definition and rules of logarithms (exponentials and change-of-base formula), you will be taught how to solve logarithmic equations.

## Related Topics

## 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.
- Plugin the answers back into the original equation and check to see if the solution works.

## Examples

### 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: \(36-x^2=16→36-16=x^2→x^2=20→x=\sqrt{20}=2\sqrt{5}\)

### 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: \(log(3x+10)=log(3(6)+10)=log(28) \)

Logarithms of negative numbers are not defined. Therefore, there is no solution for this equation.

## Exercises for Solving Logarithmic Equations

### Solve Logarithmic Equations.

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

- \(\color{blue}{\frac{1}{3}}\)
- \(\color{blue}{\frac{2}{3}}\)
- \(\color{blue}{\frac{1}{100}}\)
- \(\color{blue}{\frac{1}{2}}\)
- \(\color{blue}{2}\)
- \(\color{blue}{No \ solution \ for \ x∈R}\)