Precalculus Examples

$\log_{5} (5^{\frac{3}{4}}) - \log_{5} (5^{2})$

Step 1

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

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

Step 2

Move $5^{\frac{3}{4}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\log_{5} (\frac{1}{5^{2} \cdot 5^{- \frac{3}{4}}})$

Step 3

Multiply $5^{2}$ by $5^{- \frac{3}{4}}$ by adding the exponents.

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Step 3.1

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

$\log_{5} (\frac{1}{5^{2 - \frac{3}{4}}})$

Step 3.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\log_{5} (\frac{1}{5^{2 \cdot \frac{4}{4} - \frac{3}{4}}})$

Step 3.3

Combine $2$ and $\frac{4}{4}$ .

$\log_{5} (\frac{1}{5^{\frac{2 \cdot 4}{4} - \frac{3}{4}}})$

Step 3.4

Combine the numerators over the common denominator.

$\log_{5} (\frac{1}{5^{\frac{2 \cdot 4 - 3}{4}}})$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $2$ by $4$ .

$\log_{5} (\frac{1}{5^{\frac{8 - 3}{4}}})$

Step 3.5.2

Subtract $3$ from $8$ .

$\log_{5} (\frac{1}{5^{\frac{5}{4}}})$

Step 4

Move $5^{\frac{5}{4}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\log_{5} (5^{- \frac{5}{4}})$

Step 5

Expand $\log_{5} (5^{- \frac{5}{4}})$ by moving $- \frac{5}{4}$ outside the logarithm.

$- \frac{5}{4} \log_{5} (5)$

Step 6

Logarithm base $5$ of $5$ is $1$ .

$- \frac{5}{4} \cdot 1$

Step 7

Multiply $- 1$ by $1$ .

$- \frac{5}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{5}{4}$

Decimal Form:

$- 1.25$

Mixed Number Form:

$- 1 \frac{1}{4}$