Precalculus Examples

5 {log}_{b} (2 x) - {log}_{b} (x)

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

Step 1

Simplify each term.

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Step 1.1

Simplify

5 {log}_{b} (2 x)

$5 \log_{b} (2 x)$ by moving

5

$5$ inside the logarithm.

{log}_{b} ({(2 x)}^{5}) - {log}_{b} (x)

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

Step 1.2

Apply the product rule to

2 x

$2 x$ .

{log}_{b} (2^{5} x^{5}) - {log}_{b} (x)

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

Step 1.3

Raise

2

$2$ to the power of

5

$5$ .

{log}_{b} (32 x^{5}) - {log}_{b} (x)

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

{log}_{b} (32 x^{5}) - {log}_{b} (x)

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

Step 2

Use the quotient property of logarithms,

{log}_{b} (x) - {log}_{b} (y) = {log}_{b} (\frac{x}{y})

$\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

{log}_{b} (\frac{32 x^{5}}{x})

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

Step 3

Cancel the common factor of

x^{5}

$x^{5}$ and

x

$x$ .

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Step 3.1

Factor

x

$x$ out of

32 x^{5}

$32 x^{5}$ .

{log}_{b} (\frac{x (32 x^{4})}{x})

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Raise

x

$x$ to the power of

1

$1$ .

{log}_{b} (\frac{x (32 x^{4})}{x^{1}})

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

Step 3.2.2

Factor

x

$x$ out of

x^{1}

$x^{1}$ .

{log}_{b} (\frac{x (32 x^{4})}{x \cdot 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

$1$ .

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

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

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

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$