Precalculus Examples

$\log (10 x) = - 2$

Step 1

Rewrite $\log (10 x) = - 2$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{- 2} = 10 x$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $10 x = 10^{- 2}$ .

$10 x = 10^{- 2}$

Step 2.2

Divide each term in $10 x = 10^{- 2}$ by $10$ and simplify.

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Step 2.2.1

Divide each term in $10 x = 10^{- 2}$ by $10$ .

$\frac{10 x}{10} = \frac{10^{- 2}}{10}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $10$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{10^{- 2}}{10}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10^{- 2}}{10}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move $10^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x = \frac{1}{10 \cdot 10^{2}}$

Step 2.2.3.2

Multiply $10$ by $10^{2}$ by adding the exponents.

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Step 2.2.3.2.1

Multiply $10$ by $10^{2}$ .

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Step 2.2.3.2.1.1

Raise $10$ to the power of $1$ .

$x = \frac{1}{10^{1} \cdot 10^{2}}$

Step 2.2.3.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{1}{10^{1 + 2}}$

Step 2.2.3.2.2

Add $1$ and $2$ .

$x = \frac{1}{10^{3}}$

Step 2.2.3.3

Raise $10$ to the power of $3$ .

$x = \frac{1}{1000}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{1000}$

Decimal Form:

$x = 0.001$