Precalculus Examples

$\log_{0.24} (\frac{3}{9})$

Step 1

Cancel the common factor of $3$ and $9$ .

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Step 1.1

Factor $3$ out of $3$ .

$\log_{0.24} (\frac{3 (1)}{9})$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $3$ out of $9$ .

$\log_{0.24} (\frac{3 \cdot 1}{3 \cdot 3})$

Step 1.2.2

Cancel the common factor.

$\log_{0.24} (\frac{3 \cdot 1}{3 \cdot 3})$

Step 1.2.3

Rewrite the expression.

$\log_{0.24} (\frac{1}{3})$

Step 2

Rewrite $\log_{0.24} (\frac{1}{3})$ using the change of base formula.

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Step 2.1

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 $0$ .

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

Step 2.2

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

$\frac{\log (\frac{1}{3})}{\log (0.24)}$

Step 3

Log base $10$ of $0.24$ is approximately $- 0.61978875$ .

$\frac{\log (\frac{1}{3})}{- 0.61978875}$

Step 4

Move the negative in front of the fraction.

$- \frac{\log (\frac{1}{3})}{0.61978875}$

Step 5

Divide $1$ by $3$ .

$- \frac{\log (0.\overline{3})}{0.61978875}$

Step 6

Log base $10$ of $0.\overline{3}$ is approximately $- 0.47712125$ .

$- \frac{- 0.47712125}{0.61978875}$

Step 7

Divide $- 0.47712125$ by $0.61978875$ .

$- - 0.76981269$

Step 8

Multiply $- 1$ by $- 0.76981269$ .

$0.76981269$