Precalculus Examples

$2 \log_{5} (x^{3}) + 3 \log_{5} (x) - 6 \log_{5} (x^{- (2)})$

Step 1

Simplify each term.

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Step 1.1

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

$\log_{5} ({(x^{3})}^{2}) + 3 \log_{5} (x) - 6 \log_{5} (x^{- (2)})$

Step 1.2

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

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Step 1.2.1

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

$\log_{5} (x^{3 \cdot 2}) + 3 \log_{5} (x) - 6 \log_{5} (x^{- (2)})$

Step 1.2.2

Multiply $3$ by $2$ .

$\log_{5} (x^{6}) + 3 \log_{5} (x) - 6 \log_{5} (x^{- (2)})$

Step 1.3

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

$\log_{5} (x^{6}) + \log_{5} (x^{3}) - 6 \log_{5} (x^{- (2)})$

Step 1.4

Multiply $- 1$ by $2$ .

$\log_{5} (x^{6}) + \log_{5} (x^{3}) - 6 \log_{5} (x^{- 2})$

Step 1.5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\log_{5} (x^{6}) + \log_{5} (x^{3}) - 6 \log_{5} (\frac{1}{x^{2}})$

Step 1.6

Simplify $- 6 \log_{5} (\frac{1}{x^{2}})$ by moving $6$ inside the logarithm.

$\log_{5} (x^{6}) + \log_{5} (x^{3}) - \log_{5} ({(\frac{1}{x^{2}})}^{6})$

Step 1.7

Apply the product rule to $\frac{1}{x^{2}}$ .

$\log_{5} (x^{6}) + \log_{5} (x^{3}) - \log_{5} (\frac{1^{6}}{{(x^{2})}^{6}})$

Step 1.8

One to any power is one.

$\log_{5} (x^{6}) + \log_{5} (x^{3}) - \log_{5} (\frac{1}{{(x^{2})}^{6}})$

Step 1.9

Multiply the exponents in ${(x^{2})}^{6}$ .

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Step 1.9.1

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

$\log_{5} (x^{6}) + \log_{5} (x^{3}) - \log_{5} (\frac{1}{x^{2 \cdot 6}})$

Step 1.9.2

Multiply $2$ by $6$ .

$\log_{5} (x^{6}) + \log_{5} (x^{3}) - \log_{5} (\frac{1}{x^{12}})$

Step 2

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

$\log_{5} (x^{6} x^{3}) - \log_{5} (\frac{1}{x^{12}})$

Step 3

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

$\log_{5} (\frac{x^{6} x^{3}}{\frac{1}{x^{12}}})$

Step 4

Multiply $x^{6}$ by $x^{3}$ by adding the exponents.

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Step 4.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{5} (\frac{x^{6 + 3}}{\frac{1}{x^{12}}})$

Step 4.2

Add $6$ and $3$ .

$\log_{5} (\frac{x^{9}}{\frac{1}{x^{12}}})$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\log_{5} (x^{9} x^{12})$

Step 6

Multiply $x^{9}$ by $x^{12}$ by adding the exponents.

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Step 6.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{5} (x^{9 + 12})$

Step 6.2

Add $9$ and $12$ .

$\log_{5} (x^{21})$