Evaluate log base 0.5 of 3

$\log_{0.5} (3)$

Step 1

Rewrite $\log_{0.5} (3)$ using the change of base formula.

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 $0$ .

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

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

$\frac{\log (3)}{\log (0.5)}$

Step 2

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

$\frac{\log (3)}{- 0.30102999}$

Step 3

Move the negative in front of the fraction.

$- \frac{\log (3)}{0.30102999}$

Step 4

Log base $10$ of $3$ is approximately $0.47712125$ .

$- \frac{0.47712125}{0.30102999}$

Step 5

Divide $0.47712125$ by $0.30102999$ .

$- 1 \cdot 1.5849625$

Step 6

Multiply $- 1$ by $1.5849625$ .

$- 1.5849625$