Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite $\log_{100} (x^{2})$ using the change of base formula.

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Step 1.1.1

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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} - \log (x^{3})$

Step 1.1.2

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

$\frac{\log (x^{2})}{\log (100)} - \log (x^{3})$

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

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

$\log (x) - \log (x^{3})$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{x}{x^{3}})$

Step 3

Cancel the common factor of $x$ and $x^{3}$ .

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Step 3.1

Raise $x$ to the power of $1$ .

$\log (\frac{x^{1}}{x^{3}})$

Step 3.2

Factor $x$ out of $x^{1}$ .

$\log (\frac{x \cdot 1}{x^{3}})$

Step 3.3

Cancel the common factors.

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Step 3.3.1

Factor $x$ out of $x^{3}$ .

$\log (\frac{x \cdot 1}{x \cdot x^{2}})$

Step 3.3.2

Cancel the common factor.

$\log (\frac{x \cdot 1}{x \cdot x^{2}})$

Step 3.3.3

Rewrite the expression.

$\log (\frac{1}{x^{2}})$