Precalculus Examples

$- \frac{8}{9} \cdot \log_{5} (5 m^{2}) + \frac{1}{2} \cdot \log_{5} (25 m^{2})$

Step 1

Rewrite $\log_{5} (5 m^{2})$ as $\log_{5} (5) + \log_{5} (m^{2})$ .

$- \frac{8}{9} \cdot (\log_{5} (5) + \log_{5} (m^{2})) + \frac{1}{2} \cdot \log_{5} (25 m^{2})$

Step 2

Expand $\log_{5} (m^{2})$ by moving $2$ outside the logarithm.

$- \frac{8}{9} \cdot (\log_{5} (5) + 2 \log_{5} (m)) + \frac{1}{2} \cdot \log_{5} (25 m^{2})$

Step 3

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

$- \frac{8}{9} \cdot (1 + 2 \log_{5} (m)) + \frac{1}{2} \cdot \log_{5} (25 m^{2})$

Step 4

Rewrite $\log_{5} (25 m^{2})$ as $\log_{5} (25) + \log_{5} (m^{2})$ .

$- \frac{8}{9} \cdot (1 + 2 \log_{5} (m)) + \frac{1}{2} \cdot (\log_{5} (25) + \log_{5} (m^{2}))$

Step 5

Expand $\log_{5} (m^{2})$ by moving $2$ outside the logarithm.

$- \frac{8}{9} \cdot (1 + 2 \log_{5} (m)) + \frac{1}{2} \cdot (\log_{5} (25) + 2 \log_{5} (m))$

Step 6

Logarithm base $5$ of $25$ is $2$ .

$- \frac{8}{9} \cdot (1 + 2 \log_{5} (m)) + \frac{1}{2} \cdot (2 + 2 \log_{5} (m))$

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

$- \frac{8}{9} \cdot 1 - \frac{8}{9} (2 \log_{5} (m)) + \frac{1}{2} \cdot (2 + 2 \log_{5} (m))$

Step 7.2

Multiply $- 1$ by $1$ .

$- \frac{8}{9} - \frac{8}{9} (2 \log_{5} (m)) + \frac{1}{2} \cdot (2 + 2 \log_{5} (m))$

Step 7.3

Multiply $- \frac{8}{9} (2 \log_{5} (m))$ .

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Step 7.3.1

Multiply $2$ by $- 1$ .

$- \frac{8}{9} - 2 (\frac{8}{9}) \log_{5} (m) + \frac{1}{2} \cdot (2 + 2 \log_{5} (m))$

Step 7.3.2

Combine $- 2$ and $\frac{8}{9}$ .

$- \frac{8}{9} + \frac{- 2 \cdot 8}{9} \log_{5} (m) + \frac{1}{2} \cdot (2 + 2 \log_{5} (m))$

Step 7.3.3

Multiply $- 2$ by $8$ .

$- \frac{8}{9} + \frac{- 16}{9} \log_{5} (m) + \frac{1}{2} \cdot (2 + 2 \log_{5} (m))$

Step 7.3.4

Combine $\frac{- 16}{9}$ and $\log_{5} (m)$ .

$- \frac{8}{9} + \frac{- 16 \log_{5} (m)}{9} + \frac{1}{2} \cdot (2 + 2 \log_{5} (m))$

Step 7.4

Move the negative in front of the fraction.

$- \frac{8}{9} - \frac{16 \log_{5} (m)}{9} + \frac{1}{2} \cdot (2 + 2 \log_{5} (m))$

Step 7.5

Apply the distributive property.

$- \frac{8}{9} - \frac{16 \log_{5} (m)}{9} + \frac{1}{2} \cdot 2 + \frac{1}{2} (2 \log_{5} (m))$

Step 7.6

Cancel the common factor of $2$ .

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Step 7.6.1

Cancel the common factor.

$- \frac{8}{9} - \frac{16 \log_{5} (m)}{9} + \frac{1}{2} \cdot 2 + \frac{1}{2} (2 \log_{5} (m))$

Step 7.6.2

Rewrite the expression.

$- \frac{8}{9} - \frac{16 \log_{5} (m)}{9} + 1 + \frac{1}{2} (2 \log_{5} (m))$

Step 7.7

Cancel the common factor of $2$ .

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Step 7.7.1

Factor $2$ out of $2 \log_{5} (m)$ .

$- \frac{8}{9} - \frac{16 \log_{5} (m)}{9} + 1 + \frac{1}{2} (2 (\log_{5} (m)))$

Step 7.7.2

Cancel the common factor.

$- \frac{8}{9} - \frac{16 \log_{5} (m)}{9} + 1 + \frac{1}{2} (2 \log_{5} (m))$

Step 7.7.3

Rewrite the expression.

$- \frac{8}{9} - \frac{16 \log_{5} (m)}{9} + 1 + \log_{5} (m)$

Step 8

Write $1$ as a fraction with a common denominator.

$- \frac{8}{9} + \frac{9}{9} - \frac{16 \log_{5} (m)}{9} + \log_{5} (m)$

Step 9

Combine the numerators over the common denominator.

$\frac{- 8 + 9}{9} - \frac{16 \log_{5} (m)}{9} + \log_{5} (m)$

Step 10

Add $- 8$ and $9$ .

$\frac{1}{9} - \frac{16 \log_{5} (m)}{9} + \log_{5} (m)$

Step 11

To write $\log_{5} (m)$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{1}{9} - \frac{16 \log_{5} (m)}{9} + \log_{5} (m) \cdot \frac{9}{9}$

Step 12

Combine $\log_{5} (m)$ and $\frac{9}{9}$ .

$\frac{1}{9} - \frac{16 \log_{5} (m)}{9} + \frac{\log_{5} (m) \cdot 9}{9}$

Step 13

Combine the numerators over the common denominator.

$\frac{1}{9} + \frac{- 16 \log_{5} (m) + \log_{5} (m) \cdot 9}{9}$

Step 14

Add $- 16 \log_{5} (m)$ and $\log_{5} (m) \cdot 9$ .

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Step 14.1

Reorder $\log_{5} (m)$ and $9$ .

$\frac{1}{9} + \frac{- 16 \log_{5} (m) + 9 \cdot \log_{5} (m)}{9}$

Step 14.2

Add $- 16 \log_{5} (m)$ and $9 \cdot \log_{5} (m)$ .

$\frac{1}{9} + \frac{- 7 \log_{5} (m)}{9}$

Step 15

Move the negative in front of the fraction.

$\frac{1}{9} - \frac{7 \log_{5} (m)}{9}$