Precalculus Examples

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

Step 1

Multiply $\log_{4} (3)$ by $\log_{3} (5)$ .

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

Step 2

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

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{3} (5)$

Step 2.2

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

$\frac{\log (3)}{\log (4)} \log_{3} (5)$

Step 3

Rewrite $\log_{3} (5)$ 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 $0$ .

$\frac{\log (3)}{\log (4)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 3.2

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

$\frac{\log (3)}{\log (4)} \cdot \frac{\log (5)}{\log (3)}$

Step 4

Cancel the common factor of $\log (3)$ .

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Step 4.1

Cancel the common factor.

$\frac{\log (3)}{\log (4)} \cdot \frac{\log (5)}{\log (3)}$

Step 4.2

Rewrite the expression.

$\frac{1}{\log (4)} \log (5)$

Step 5

Combine $\frac{1}{\log (4)}$ and $\log (5)$ .

$\frac{\log (5)}{\log (4)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (5)}{\log (4)}$

Decimal Form:

$1.16096404 \dots$