Precalculus Examples

$e^{\ln (7) + 2 \ln (4) - \log_{9} (\frac{\sqrt{27}}{27})}$

Step 1

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$e^{\ln (7) + 2 \ln (2^{2}) - \log_{9} (\frac{\sqrt{27}}{27})}$

Step 2

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

$e^{\ln (7) + 2 (2 \ln (2)) - \log_{9} (\frac{\sqrt{27}}{27})}$

Step 3

Multiply $2$ by $2$ .

$e^{\ln (7) + 4 \ln (2) - \log_{9} (\frac{\sqrt{27}}{27})}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 4.1.1

Factor $9$ out of $27$ .

$e^{\ln (7) + 4 \ln (2) - \log_{9} (\frac{\sqrt{9 (3)}}{27})}$

Step 4.1.2

Rewrite $9$ as $3^{2}$ .

$e^{\ln (7) + 4 \ln (2) - \log_{9} (\frac{\sqrt{3^{2} \cdot 3}}{27})}$

Step 4.2

Pull terms out from under the radical.

$e^{\ln (7) + 4 \ln (2) - \log_{9} (\frac{3 \sqrt{3}}{27})}$

Step 5

Reduce the expression $\frac{3 \sqrt{3}}{27}$ by cancelling the common factors.

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Step 5.1

Factor $3$ out of $3 \sqrt{3}$ .

$e^{\ln (7) + 4 \ln (2) - \log_{9} (\frac{3 (\sqrt{3})}{27})}$

Step 5.2

Factor $3$ out of $27$ .

$e^{\ln (7) + 4 \ln (2) - \log_{9} (\frac{3 \sqrt{3}}{3 \cdot 9})}$

Step 5.3

Cancel the common factor.

$e^{\ln (7) + 4 \ln (2) - \log_{9} (\frac{3 \sqrt{3}}{3 \cdot 9})}$

Step 5.4

Rewrite the expression.

$e^{\ln (7) + 4 \ln (2) - \log_{9} (\frac{\sqrt{3}}{9})}$

Step 6

Rewrite $\frac{\sqrt{3}}{9}$ as $\sqrt{3} \cdot 9^{- 1}$ .

$e^{\ln (7) + 4 \ln (2) - \log_{9} (\sqrt{3} \cdot 9^{- 1})}$

Step 7

Rewrite $\log_{9} (\sqrt{3} \cdot 9^{- 1})$ as $\log_{9} (\sqrt{3}) + \log_{9} (9^{- 1})$ .

$e^{\ln (7) + 4 \ln (2) - (\log_{9} (\sqrt{3}) + \log_{9} (9^{- 1}))}$

Step 8

Use logarithm rules to move $- 1$ out of the exponent.

$e^{\ln (7) + 4 \ln (2) - (\log_{9} (\sqrt{3}) - \log_{9} (9))}$

Step 9

Logarithm base $9$ of $9$ is $1$ .

$e^{\ln (7) + 4 \ln (2) - (\log_{9} (\sqrt{3}) - 1 \cdot 1)}$

Step 10

Multiply $- 1$ by $1$ .

$e^{\ln (7) + 4 \ln (2) - (\log_{9} (\sqrt{3}) - 1)}$

Step 11

Apply the distributive property.

$e^{\ln (7) + 4 \ln (2) - \log_{9} (\sqrt{3}) - - 1}$

Step 12

Multiply $- 1$ by $- 1$ .

$e^{\ln (7) + 4 \ln (2) - \log_{9} (\sqrt{3}) + 1}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$e^{\ln (7) + 4 \ln (2) - \log_{9} (\sqrt{3}) + 1}$

Decimal Form:

$237.10400186 \dots$