Precalculus Examples

$\ln (\frac{\frac{9}{2}}{\ln (3)} + 3) \cdot \frac{1}{5}$

Step 1

Simplify each term.

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Step 1.1

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{9}{2} \cdot \frac{1}{\ln (3)} + 3) \cdot \frac{1}{5}$

Step 1.2

Multiply $\frac{9}{2} \cdot \frac{1}{\ln (3)}$ .

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Step 1.2.1

Multiply $\frac{9}{2}$ by $\frac{1}{\ln (3)}$ .

$\ln (\frac{9}{2 \ln (3)} + 3) \cdot \frac{1}{5}$

Step 1.2.2

Simplify $2 \ln (3)$ by moving $2$ inside the logarithm.

$\ln (\frac{9}{\ln (3^{2})} + 3) \cdot \frac{1}{5}$

Step 1.3

Raise $3$ to the power of $2$ .

$\ln (\frac{9}{\ln (9)} + 3) \cdot \frac{1}{5}$

Step 2

To write $3$ as a fraction with a common denominator, multiply by $\frac{\ln (9)}{\ln (9)}$ .

$\ln (\frac{9}{\ln (9)} + \frac{3 \ln (9)}{\ln (9)}) \cdot \frac{1}{5}$

Step 3

Combine the numerators over the common denominator.

$\ln (\frac{9 + 3 \ln (9)}{\ln (9)}) \cdot \frac{1}{5}$

Step 4

Simplify the numerator.

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Step 4.1

Simplify $3 \ln (9)$ by moving $3$ inside the logarithm.

$\ln (\frac{9 + \ln (9^{3})}{\ln (9)}) \cdot \frac{1}{5}$

Step 4.2

Raise $9$ to the power of $3$ .

$\ln (\frac{9 + \ln (729)}{\ln (9)}) \cdot \frac{1}{5}$

Step 5

Multiply $\ln (\frac{9 + \ln (729)}{\ln (9)}) \frac{1}{5}$ .

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Step 5.1

Reorder $\ln (\frac{9 + \ln (729)}{\ln (9)})$ and $\frac{1}{5}$ .

$\frac{1}{5} \ln (\frac{9 + \ln (729)}{\ln (9)})$

Step 5.2

Simplify $\frac{1}{5} \ln (\frac{9 + \ln (729)}{\ln (9)})$ by moving $\frac{1}{5}$ inside the logarithm.

$\ln ({(\frac{9 + \ln (729)}{\ln (9)})}^{\frac{1}{5}})$

Step 6

Apply the product rule to $\frac{9 + \ln (729)}{\ln (9)}$ .

$\ln (\frac{{(9 + \ln (729))}^{\frac{1}{5}}}{{\ln (9)}^{\frac{1}{5}}})$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\ln (\frac{{(9 + \ln (729))}^{\frac{1}{5}}}{{(\ln (9))}^{\frac{1}{5}}})$

Decimal Form:

$0.39190840 \dots$