Precalculus Examples

$3 \log_{5} (x) + 4 \log_{5} (y) - \log_{5} (z) - 2 \log_{5} (4)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log_{5} (x)$ by moving $3$ inside the logarithm.

$\log_{5} (x^{3}) + 4 \log_{5} (y) - \log_{5} (z) - 2 \log_{5} (4)$

Step 1.2

Simplify $4 \log_{5} (y)$ by moving $4$ inside the logarithm.

$\log_{5} (x^{3}) + \log_{5} (y^{4}) - \log_{5} (z) - 2 \log_{5} (4)$

Step 1.3

Simplify $- 2 \log_{5} (4)$ by moving $2$ inside the logarithm.

$\log_{5} (x^{3}) + \log_{5} (y^{4}) - \log_{5} (z) - \log_{5} (4^{2})$

Step 1.4

Raise $4$ to the power of $2$ .

$\log_{5} (x^{3}) + \log_{5} (y^{4}) - \log_{5} (z) - \log_{5} (16)$

Step 2

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

$\log_{5} (x^{3} y^{4}) - \log_{5} (z) - \log_{5} (16)$

Step 3

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

$\log_{5} (\frac{x^{3} y^{4}}{z}) - \log_{5} (16)$

Step 4

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

$\log_{5} (\frac{\frac{x^{3} y^{4}}{z}}{16})$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\log_{5} (\frac{x^{3} y^{4}}{z} \cdot \frac{1}{16})$

Step 6

Combine.

$\log_{5} (\frac{x^{3} y^{4} \cdot 1}{z \cdot 16})$

Step 7

Simplify the expression.

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Step 7.1

Multiply $x^{3}$ by $1$ .

$\log_{5} (\frac{x^{3} y^{4}}{z \cdot 16})$

Step 7.2

Move $16$ to the left of $z$ .

$\log_{5} (\frac{x^{3} y^{4}}{16 z})$