Precalculus Examples

$\log (100) + \log_{x} (\sqrt{x}) - \log_{2} (2^{3})$

Step 1

Simplify each term.

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Step 1.1

Logarithm base $10$ of $100$ is $2$ .

$2 + \log_{x} (\sqrt{x}) - \log_{2} (2^{3})$

Step 1.2

Use logarithm rules to move $3$ out of the exponent.

$2 + \log_{x} (\sqrt{x}) - (3 \log_{2} (2))$

Step 1.3

Logarithm base $2$ of $2$ is $1$ .

$2 + \log_{x} (\sqrt{x}) - (3 \cdot 1)$

Step 1.4

Multiply $3$ by $1$ .

$2 + \log_{x} (\sqrt{x}) - 1 \cdot 3$

Step 1.5

Multiply $- 1$ by $3$ .

$2 + \log_{x} (\sqrt{x}) - 3$

Step 2

Subtract $3$ from $2$ .

$- 1 + \log_{x} (\sqrt{x})$

Step 3

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

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Step 3.1

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 $0$ .

$- 1 + \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 3.2

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

$- 1 + \frac{\log (\sqrt{x})}{\log (x)}$