Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 4

Multiply $x^{6}$ by $x^{- \frac{1}{2}}$ 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.

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

Step 4.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.3

Combine $6$ and $\frac{2}{2}$ .

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $6$ by $2$ .

$\log_{5} (\frac{1}{x^{\frac{12 - 1}{2}}})$

Step 4.5.2

Subtract $1$ from $12$ .

$\log_{5} (\frac{1}{x^{\frac{11}{2}}})$