Precalculus Examples

$(\log_{9} (96)) - (5 \log_{9} (2)) - 4$

Step 1

Simplify each term.

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Step 1.1

Simplify $5 \log_{9} (2)$ by moving $5$ inside the logarithm.

$\log_{9} (96) - \log_{9} (2^{5}) - 4$

Step 1.2

Raise $2$ to the power of $5$ .

$\log_{9} (96) - \log_{9} (32) - 4$

Step 2

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

$\log_{9} (\frac{96}{32}) - 4$

Step 3

Logarithm base $9$ of $\frac{96}{32}$ is $\frac{1}{2}$ .

$\frac{1}{2} - 4$

Step 4

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

$\frac{1}{2} - 4 \cdot \frac{2}{2}$

Step 5

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

$\frac{1}{2} + \frac{- 4 \cdot 2}{2}$

Step 6

Combine the numerators over the common denominator.

$\frac{1 - 4 \cdot 2}{2}$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 4$ by $2$ .

$\frac{1 - 8}{2}$

Step 7.2

Subtract $8$ from $1$ .

$\frac{- 7}{2}$

Step 8

Move the negative in front of the fraction.

$- \frac{7}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{7}{2}$

Decimal Form:

$- 3.5$

Mixed Number Form:

$- 3 \frac{1}{2}$