Precalculus Examples

$\frac{\log_{18} (5) \cdot 25}{36}$

Step 1

Simplify the numerator.

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Step 1.1

Reorder $\log_{18} (5)$ and $25$ .

$\frac{25 \cdot \log_{18} (5)}{36}$

Step 1.2

Simplify $25 \cdot \log_{18} (5)$ by moving $25$ inside the logarithm.

$\frac{\log_{18} (5^{25})}{36}$

Step 2

Raise $5$ to the power of $25$ .

$\frac{\log_{18} (298023223876953000)}{36}$

Step 3

Rewrite $\frac{\log_{18} (298023223876953000)}{36}$ as $\frac{1}{36} \log_{18} (298023223876953000)$ .

$\frac{1}{36} \log_{18} (298023223876953000)$

Step 4

Simplify $\frac{1}{36} \log_{18} (298023223876953000)$ by moving $\frac{1}{36}$ inside the logarithm.

$\log_{18} (298023223876953000^{\frac{1}{36}})$

Step 5

Rewrite $\log_{18} (298023223876953000^{\frac{1}{36}})$ using the change of base formula.

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Step 5.1

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 5.2

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

$\frac{\log (298023223876953000^{\frac{1}{36}})}{\log (18)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (298023223876953000^{\frac{1}{36}})}{\log (18)}$

Decimal Form:

$0.38668562 \dots$