Precalculus Examples

$3 (\log_{b} (x) - 3 \log_{b} (z)) + 3 \log_{b} (y)$

Step 1

Simplify each term.

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Step 1.1

Simplify $- 3 \log_{b} (z)$ by moving $3$ inside the logarithm.

$3 (\log_{b} (x) - \log_{b} (z^{3})) + 3 \log_{b} (y)$

Step 1.2

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

$3 \log_{b} (\frac{x}{z^{3}}) + 3 \log_{b} (y)$

Step 1.3

Simplify $3 \log_{b} (\frac{x}{z^{3}})$ by moving $3$ inside the logarithm.

$\log_{b} ({(\frac{x}{z^{3}})}^{3}) + 3 \log_{b} (y)$

Step 1.4

Apply the product rule to $\frac{x}{z^{3}}$ .

$\log_{b} (\frac{x^{3}}{{(z^{3})}^{3}}) + 3 \log_{b} (y)$

Step 1.5

Multiply the exponents in ${(z^{3})}^{3}$ .

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Step 1.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{b} (\frac{x^{3}}{z^{3 \cdot 3}}) + 3 \log_{b} (y)$

Step 1.5.2

Multiply $3$ by $3$ .

$\log_{b} (\frac{x^{3}}{z^{9}}) + 3 \log_{b} (y)$

Step 1.6

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

$\log_{b} (\frac{x^{3}}{z^{9}}) + \log_{b} (y^{3})$

Step 2

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

$\log_{b} (\frac{x^{3}}{z^{9}} y^{3})$

Step 3

Combine $\frac{x^{3}}{z^{9}}$ and $y^{3}$ .

$\log_{b} (\frac{x^{3} y^{3}}{z^{9}})$