Precalculus Examples

$8 \log_{4} (x) - 9 \log_{4} (y) + 3 \log_{4} (3 z)$

Step 1

Simplify each term.

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Step 1.1

Simplify $8 \log_{4} (x)$ by moving $8$ inside the logarithm.

$\log_{4} (x^{8}) - 9 \log_{4} (y) + 3 \log_{4} (3 z)$

Step 1.2

Simplify $- 9 \log_{4} (y)$ by moving $9$ inside the logarithm.

$\log_{4} (x^{8}) - \log_{4} (y^{9}) + 3 \log_{4} (3 z)$

Step 1.3

Simplify $3 \log_{4} (3 z)$ by moving $3$ inside the logarithm.

$\log_{4} (x^{8}) - \log_{4} (y^{9}) + \log_{4} ({(3 z)}^{3})$

Step 1.4

Apply the product rule to $3 z$ .

$\log_{4} (x^{8}) - \log_{4} (y^{9}) + \log_{4} (3^{3} z^{3})$

Step 1.5

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

$\log_{4} (x^{8}) - \log_{4} (y^{9}) + \log_{4} (27 z^{3})$

Step 2

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

$\log_{4} (\frac{x^{8}}{y^{9}}) + \log_{4} (27 z^{3})$

Step 3

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

$\log_{4} (\frac{x^{8}}{y^{9}} (27 z^{3}))$

Step 4

Rewrite using the commutative property of multiplication.

$\log_{4} (27 \frac{x^{8}}{y^{9}} z^{3})$

Step 5

Combine $27$ and $\frac{x^{8}}{y^{9}}$ .

$\log_{4} (\frac{27 x^{8}}{y^{9}} z^{3})$

Step 6

Combine $\frac{27 x^{8}}{y^{9}}$ and $z^{3}$ .

$\log_{4} (\frac{27 x^{8} z^{3}}{y^{9}})$