Precalculus Examples

$\log_{0.125} (x)$

Step 1

Rewrite $\log_{0.125} (x)$ using the change of base formula.

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Step 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 $0$ .

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

Step 1.2

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

$\frac{\log (x)}{\log (0.125)}$

Step 2

Log base $10$ of $0.125$ is approximately $- 0.90308998$ .

$\frac{\log (x)}{- 0.90308998}$

Step 3

Move the negative in front of the fraction.

$- \frac{\log (x)}{0.90308998}$

Step 4

Multiply by $1$ .

$- \frac{1 \log (x)}{0.90308998}$

Step 5

Factor $0.90308998$ out of $0.90308998$ .

$- \frac{1 \log (x)}{0.90308998 (1)}$

Step 6

Separate fractions.

$- (\frac{1}{0.90308998} \cdot \frac{\log (x)}{1})$

Step 7

Divide $1$ by $0.90308998$ .

$- (1.10730936 \frac{\log (x)}{1})$

Step 8

Divide $\log (x)$ by $1$ .

$- (1.10730936 \log (x))$

Step 9

Simplify $1.10730936 \log (x)$ by moving $1.10730936$ inside the logarithm.

$- \log (x^{1.10730936})$