Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

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

$2 \log_{5} (\frac{x y^{2}}{z^{3}})$

Step 4

Simplify $2 \log_{5} (\frac{x y^{2}}{z^{3}})$ by moving $2$ inside the logarithm.

$\log_{5} ({(\frac{x y^{2}}{z^{3}})}^{2})$

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1

Apply the product rule to $\frac{x y^{2}}{z^{3}}$ .

$\log_{5} (\frac{{(x y^{2})}^{2}}{{(z^{3})}^{2}})$

Step 5.2

Apply the product rule to $x y^{2}$ .

$\log_{5} (\frac{x^{2} {(y^{2})}^{2}}{{(z^{3})}^{2}})$

Step 6

Multiply the exponents in ${(y^{2})}^{2}$ .

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Step 6.1

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

$\log_{5} (\frac{x^{2} y^{2 \cdot 2}}{{(z^{3})}^{2}})$

Step 6.2

Multiply $2$ by $2$ .

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

Step 7

Multiply the exponents in ${(z^{3})}^{2}$ .

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Step 7.1

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

$\log_{5} (\frac{x^{2} y^{4}}{z^{3 \cdot 2}})$

Step 7.2

Multiply $3$ by $2$ .

$\log_{5} (\frac{x^{2} y^{4}}{z^{6}})$

Step 8

Rewrite $\log_{5} (\frac{x^{2} y^{4}}{z^{6}})$ using the change of base formula.

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Step 8.1

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 $0$ .

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

Step 8.2

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

$\frac{\log (\frac{x^{2} y^{4}}{z^{6}})}{\log (5)}$