Algebra Examples

$\log_{0.1} (8)$

Step 1

Rewrite $\log_{0.1} (8)$ 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 $b = 10$ .

$\frac{\log (8)}{\log (0.1)}$

Step 2

Log base $10$ of $0.1$ is approximately $- 1$ .

$\frac{\log (8)}{- 1}$

Step 3

Rewrite $\log (8)$ as $\log (2^{3})$ .

$\frac{\log (2^{3})}{- 1}$

Step 4

Expand $\log (2^{3})$ by moving $3$ outside the logarithm.

$\frac{3 \log (2)}{- 1}$

Step 5

Cancel the common factor of $3$ and $- 1$ .

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Step 5.1

Factor $1$ out of $3 \log (2)$ .

$\frac{1 (3 \log (2))}{- 1}$

Step 5.2

Move the negative one from the denominator of $\frac{3 \log (2)}{- 1}$ .

$- 1 \cdot (3 \log (2))$

Step 6

Simplify $3 \log (2)$ by moving $3$ inside the logarithm.

$- 1 \cdot \log (2^{3})$

Step 7

Raise $2$ to the power of $3$ .

$- 1 \cdot \log (8)$

Step 8

Rewrite $- 1 \log (8)$ as $- \log (8)$ .

$- \log (8)$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \log (8)$

Decimal Form:

$- 0.90308998 \dots$