Precalculus Examples

$\log (x) - \log (6) = 3$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{x}{6}) = 3$

Step 2

Rewrite $\log (\frac{x}{6}) = 3$ 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 $b^{y} = x$ .

$10^{3} = \frac{x}{6}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $\frac{x}{6} = 10^{3}$ .

$\frac{x}{6} = 10^{3}$

Step 3.2

Multiply both sides of the equation by $6$ .

$6 \frac{x}{6} = 6 \cdot 10^{3}$

Step 3.3

Simplify the left side.

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Step 3.3.1

Cancel the common factor of $6$ .

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Step 3.3.1.1

Cancel the common factor.

$6 \frac{x}{6} = 6 \cdot 10^{3}$

Step 3.3.1.2

Rewrite the expression.

$x = 6 \cdot 10^{3}$

Step 4

The result can be shown in multiple forms.

Scientific Notation:

$x = 6 \cdot 10^{3}$

Expanded Form:

$x = 6000$