Precalculus Examples

$5 \log_{b} (2 x) - \log_{b} (x)$

Step 1

Simplify each term.

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Step 1.1

Simplify $5 \log_{b} (2 x)$ by moving $5$ inside the logarithm.

$\log_{b} ({(2 x)}^{5}) - \log_{b} (x)$

Step 1.2

Apply the product rule to $2 x$ .

$\log_{b} (2^{5} x^{5}) - \log_{b} (x)$

Step 1.3

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

$\log_{b} (32 x^{5}) - \log_{b} (x)$

Step 2

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

$\log_{b} (\frac{32 x^{5}}{x})$

Step 3

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

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Step 3.1

Factor $x$ out of $32 x^{5}$ .

$\log_{b} (\frac{x (32 x^{4})}{x})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

$\log_{b} (\frac{x (32 x^{4})}{x^{1}})$

Step 3.2.2

Factor $x$ out of $x^{1}$ .

$\log_{b} (\frac{x (32 x^{4})}{x \cdot 1})$

Step 3.2.3

Cancel the common factor.

$\log_{b} (\frac{x (32 x^{4})}{x \cdot 1})$

Step 3.2.4

Rewrite the expression.

$\log_{b} (\frac{32 x^{4}}{1})$

Step 3.2.5

Divide $32 x^{4}$ by $1$ .

$\log_{b} (32 x^{4})$