Precalculus Examples

$\log_{5} (x + 4) + \log_{5} (8) = \log_{5} (64)$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{5} ((x + 4) \cdot 8) = \log_{5} (64)$

Step 1.2

Apply the distributive property.

$\log_{5} (x \cdot 8 + 4 \cdot 8) = \log_{5} (64)$

Step 1.3

Simplify the expression.

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Step 1.3.1

Move $8$ to the left of $x$ .

$\log_{5} (8 \cdot x + 4 \cdot 8) = \log_{5} (64)$

Step 1.3.2

Multiply $4$ by $8$ .

$\log_{5} (8 x + 32) = \log_{5} (64)$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$8 x + 32 = 64$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1.1

Subtract $32$ from both sides of the equation.

$8 x = 64 - 32$

Step 3.1.2

Subtract $32$ from $64$ .

$8 x = 32$

Step 3.2

Divide each term in $8 x = 32$ by $8$ and simplify.

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Step 3.2.1

Divide each term in $8 x = 32$ by $8$ .

$\frac{8 x}{8} = \frac{32}{8}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $8$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{32}{8}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{32}{8}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $32$ by $8$ .

$x = 4$