Precalculus Examples

$3 \log_{3} (5) - \log_{3} (4)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log_{3} (5)$ by moving $3$ inside the logarithm.

$\log_{3} (5^{3}) - \log_{3} (4)$

Step 1.2

Raise $5$ to the power of $3$ .

$\log_{3} (125) - \log_{3} (4)$

Step 2

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

$\log_{3} (\frac{125}{4})$

Step 3

Rewrite $\log_{3} (\frac{125}{4})$ 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 $0$ .

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

Step 3.2

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

$\frac{\log (\frac{125}{4})}{\log (3)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\frac{125}{4})}{\log (3)}$

Decimal Form:

$3.13306105 \dots$