Precalculus Examples

$\log_{5} (5 x) = 3$

Step 1

Rewrite $\log_{5} (5 x) = 3$ 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 $b^{y} = x$ .

$5^{3} = 5 x$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $5 x = 5^{3}$ .

$5 x = 5^{3}$

Step 2.2

Divide each term in $5 x = 5^{3}$ by $5$ and simplify.

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Step 2.2.1

Divide each term in $5 x = 5^{3}$ by $5$ .

$\frac{5 x}{5} = \frac{5^{3}}{5}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $5$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{5^{3}}{5}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5^{3}}{5}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Cancel the common factor of $5^{3}$ and $5$ .

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Step 2.2.3.1.1

Factor $5$ out of $5^{3}$ .

$x = \frac{5 \cdot 5^{2}}{5}$

Step 2.2.3.1.2

Cancel the common factors.

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Step 2.2.3.1.2.1

Factor $5$ out of $5$ .

$x = \frac{5 \cdot 5^{2}}{5 (1)}$

Step 2.2.3.1.2.2

Cancel the common factor.

$x = \frac{5 \cdot 5^{2}}{5 \cdot 1}$

Step 2.2.3.1.2.3

Rewrite the expression.

$x = \frac{5^{2}}{1}$

Step 2.2.3.1.2.4

Divide $5^{2}$ by $1$ .

$x = 5^{2}$

Step 2.2.3.2

Raise $5$ to the power of $2$ .

$x = 25$