Precalculus Examples

$\log_{b} (5 x) + 7 (\log_{b} (x) - \log_{b} (y))$

Step 1

Simplify each term.

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Step 1.1

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

$\log_{b} (5 x) + 7 \log_{b} (\frac{x}{y})$

Step 1.2

Simplify $7 \log_{b} (\frac{x}{y})$ by moving $7$ inside the logarithm.

$\log_{b} (5 x) + \log_{b} ({(\frac{x}{y})}^{7})$

Step 1.3

Apply the product rule to $\frac{x}{y}$ .

$\log_{b} (5 x) + \log_{b} (\frac{x^{7}}{y^{7}})$

Step 2

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

$\log_{b} (5 x \frac{x^{7}}{y^{7}})$

Step 3

Multiply $5 x \frac{x^{7}}{y^{7}}$ .

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Step 3.1

Combine $5$ and $\frac{x^{7}}{y^{7}}$ .

$\log_{b} (x \frac{5 x^{7}}{y^{7}})$

Step 3.2

Combine $x$ and $\frac{5 x^{7}}{y^{7}}$ .

$\log_{b} (\frac{x (5 x^{7})}{y^{7}})$

Step 3.3

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

$\log_{b} (\frac{5 (x^{1} x^{7})}{y^{7}})$

Step 3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{b} (\frac{5 x^{1 + 7}}{y^{7}})$

Step 3.5

Add $1$ and $7$ .

$\log_{b} (\frac{5 x^{8}}{y^{7}})$