Precalculus Examples

$6 \log (3 y) - 2 \log (x) + 5 \log (z)$

Step 1

Simplify each term.

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Step 1.1

Simplify $6 \log (3 y)$ by moving $6$ inside the logarithm.

$\log ({(3 y)}^{6}) - 2 \log (x) + 5 \log (z)$

Step 1.2

Apply the product rule to $3 y$ .

$\log (3^{6} y^{6}) - 2 \log (x) + 5 \log (z)$

Step 1.3

Raise $3$ to the power of $6$ .

$\log (729 y^{6}) - 2 \log (x) + 5 \log (z)$

Step 1.4

Simplify $- 2 \log (x)$ by moving $2$ inside the logarithm.

$\log (729 y^{6}) - \log (x^{2}) + 5 \log (z)$

Step 1.5

Simplify $5 \log (z)$ by moving $5$ inside the logarithm.

$\log (729 y^{6}) - \log (x^{2}) + \log (z^{5})$

Step 2

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

$\log (\frac{729 y^{6}}{x^{2}}) + \log (z^{5})$

Step 3

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

$\log (\frac{729 y^{6}}{x^{2}} z^{5})$

Step 4

Combine $\frac{729 y^{6}}{x^{2}}$ and $z^{5}$ .

$\log (\frac{729 y^{6} z^{5}}{x^{2}})$