Precalculus Examples

$\log_{8} (5) + \log_{8} (x) = 1$

Step 1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{8} (5 x) = 1$

Step 2

Rewrite $\log_{8} (5 x) = 1$ 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 $b^{y} = x$ .

$8^{1} = 5 x$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $5 x = 8^{1}$ .

$5 x = 8$

Step 3.2

Divide each term in $5 x = 8^{1}$ by $5$ and simplify.

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Step 3.2.1

Divide each term in $5 x = 8^{1}$ by $5$ .

$\frac{5 x}{5} = \frac{8^{1}}{5}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{8^{1}}{5}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{8^{1}}{5}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Evaluate the exponent.

$x = \frac{8}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{8}{5}$

Decimal Form:

$x = 1.6$

Mixed Number Form:

$x = 1 \frac{3}{5}$