Algebra Examples

$10^{2 x} = 0.1$

Step 1

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{2 x}) = \log (0.1)$

Step 2

Expand the left side.

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Step 2.1

Expand $\log (10^{2 x})$ by moving $2 x$ outside the logarithm.

$2 x \log (10) = \log (0.1)$

Step 2.2

Logarithm base $10$ of $10$ is $1$ .

$2 x \cdot 1 = \log (0.1)$

Step 2.3

Multiply $2$ by $1$ .

$2 x = \log (0.1)$

Step 3

Simplify the right side.

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Step 3.1

Log base $10$ of $0.1$ is approximately $- 1$ .

$2 x = - 1$

Step 4

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of $- 1$ and $2$ .

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Step 4.3.1.1

Rewrite $- 1$ as $1 (- 1)$ .

$x = \frac{1 (- 1)}{2}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Rewrite $2$ as $1 (2)$ .

$x = \frac{1 \cdot - 1}{1 \cdot 2}$

Step 4.3.1.2.2

Cancel the common factor.

$x = \frac{1 \cdot - 1}{1 \cdot 2}$

Step 4.3.1.2.3

Rewrite the expression.

$x = \frac{- 1}{2}$

Step 4.3.2

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{2}$

Decimal Form:

$x = - 0.5$