# How to Solve Logarithmic Equations

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 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.

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. $$log (4k-5)=log (2k-1)$$

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

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