Precalculus Examples

$\log_{a} (6 x) + 3 (\log_{a} (x) - \log_{a} (y))$

Step 1

Simplify each term.

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Step 1.1

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

$\log_{a} (6 x) + 3 \log_{a} (\frac{x}{y})$

Step 1.2

Simplify $3 \log_{a} (\frac{x}{y})$ by moving $3$ inside the logarithm.

$\log_{a} (6 x) + \log_{a} ({(\frac{x}{y})}^{3})$

Step 1.3

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

$\log_{a} (6 x) + \log_{a} (\frac{x^{3}}{y^{3}})$

Step 2

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

$\log_{a} (6 x \frac{x^{3}}{y^{3}})$

Step 3

Multiply $6 x \frac{x^{3}}{y^{3}}$ .

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Step 3.1

Combine $6$ and $\frac{x^{3}}{y^{3}}$ .

$\log_{a} (x \frac{6 x^{3}}{y^{3}})$

Step 3.2

Combine $x$ and $\frac{6 x^{3}}{y^{3}}$ .

$\log_{a} (\frac{x (6 x^{3})}{y^{3}})$

Step 3.3

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

$\log_{a} (\frac{6 (x^{1} x^{3})}{y^{3}})$

Step 3.4

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

$\log_{a} (\frac{6 x^{1 + 3}}{y^{3}})$

Step 3.5

Add $1$ and $3$ .

$\log_{a} (\frac{6 x^{4}}{y^{3}})$