Precalculus Examples

$\frac{1}{2 \log_{5} (x) - 4 \log_{5} (y) - \log_{5} (z)}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{1}{\log_{5} (x^{2}) - 4 \log_{5} (y) - \log_{5} (z)}$

Step 1.2

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

$\frac{1}{\log_{5} (x^{2}) - \log_{5} (y^{4}) - \log_{5} (z)}$

Step 1.3

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

$\frac{1}{\log_{5} (\frac{x^{2}}{y^{4}}) - \log_{5} (z)}$

Step 1.4

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

$\frac{1}{\log_{5} (\frac{\frac{x^{2}}{y^{4}}}{z})}$

Step 1.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{\log_{5} (\frac{x^{2}}{y^{4}} \cdot \frac{1}{z})}$

Step 1.6

Combine.

$\frac{1}{\log_{5} (\frac{x^{2} \cdot 1}{y^{4} z})}$

Step 1.7

Multiply $x^{2}$ by $1$ .

$\frac{1}{\log_{5} (\frac{x^{2}}{y^{4} z})}$

Step 2

Rewrite $\log_{5} (\frac{x^{2}}{y^{4} z})$ using the change of base formula.

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Step 2.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{1}{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}$

Step 2.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{1}{\frac{\log (\frac{x^{2}}{y^{4} z})}{\log (5)}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{\log (5)}{\log (\frac{x^{2}}{y^{4} z})}$

Step 4

Multiply $\frac{\log (5)}{\log (\frac{x^{2}}{y^{4} z})}$ by $1$ .

$\frac{\log (5)}{\log (\frac{x^{2}}{y^{4} z})}$