Precalculus Examples

$\log_{0.5} (16) - \log_{0.6} (\frac{1}{36})$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\log_{0.5} (16)$ using the change of base formula.

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Step 1.1.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} - \log_{0.6} (\frac{1}{36})$

Step 1.1.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (16)}{\log (0.5)} - \log_{0.6} (\frac{1}{36})$

Step 1.2

Log base $10$ of $0.5$ is approximately $- 0.30102999$ .

$\frac{\log (16)}{- 0.30102999} - \log_{0.6} (\frac{1}{36})$

Step 1.3

Move the negative in front of the fraction.

$- \frac{\log (16)}{0.30102999} - \log_{0.6} (\frac{1}{36})$

Step 1.4

Log base $10$ of $16$ is approximately $1.20411998$ .

$- \frac{1.20411998}{0.30102999} - \log_{0.6} (\frac{1}{36})$

Step 1.5

Divide $1.20411998$ by $0.30102999$ .

$- 1 \cdot 4 - \log_{0.6} (\frac{1}{36})$

Step 1.6

Multiply $- 1$ by $4$ .

$- 4 - \log_{0.6} (\frac{1}{36})$

Step 1.7

Rewrite $\log_{0.6} (\frac{1}{36})$ using the change of base formula.

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Step 1.7.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$- 4 - \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 1.7.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$- 4 - \frac{\log (\frac{1}{36})}{\log (0.6)}$

Step 1.8

Log base $10$ of $0.6$ is approximately $- 0.22184874$ .

$- 4 - \frac{\log (\frac{1}{36})}{- 0.22184874}$

Step 1.9

Move the negative in front of the fraction.

$- 4 - - \frac{\log (\frac{1}{36})}{0.22184874}$

Step 1.10

Divide $1$ by $36$ .

$- 4 - - \frac{\log (0.02\overline{7})}{0.22184874}$

Step 1.11

Log base $10$ of $0.02\overline{7}$ is approximately $- 1.5563025$ .

$- 4 - - \frac{- 1.5563025}{0.22184874}$

Step 1.12

Divide $- 1.5563025$ by $0.22184874$ .

$- 4 - - - 7.0151511$

Step 1.13

Multiply $- - - 7.0151511$ .

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Step 1.13.1

Multiply $- 1$ by $- 7.0151511$ .

$- 4 - 1 \cdot 7.0151511$

Step 1.13.2

Multiply $- 1$ by $7.0151511$ .

$- 4 - 7.0151511$

Step 2

Subtract $7.0151511$ from $- 4$ .

$- 11.0151511$