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

In this blog post, you will learn how to solve Logarithmic Equations using the properties of logarithms in a few easy steps.

## 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.

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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

### Find the value of the variables in each equation.

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

1. $$\color{blue}{95}$$
2. $$\color{blue}{4000}$$
3. $$\color{blue}{5000}$$
4. $$\color{blue}{1}$$
5. $$\color{blue}{6}$$
6. $$\color{blue}{83}$$
7. $$\color{blue}{1}$$
8. $$\color{blue}{2}$$

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