Precalculus Examples

$\log_{2} (0.04) + 2 \log_{2} (5)$

Step 1

Simplify each term.

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Step 1.1

Log base $2$ of $0.04$ is approximately $- 4.64385618$ .

$- 4.64385618 + 2 \log_{2} (5)$

Step 1.2

Simplify $2 \log_{2} (5)$ by moving $2$ inside the logarithm.

$- 4.64385618 + \log_{2} (5^{2})$

Step 1.3

Raise $5$ to the power of $2$ .

$- 4.64385618 + \log_{2} (25)$

Step 2

Rewrite $\log_{2} (25)$ 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 $0$ .

$- 4.64385618 + \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 2.2

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

$- 4.64385618 + \frac{\log (25)}{\log (2)}$

Step 3

Add $- 4.64385618$ and $\frac{\log (25)}{\log (2)}$ .

$0$