Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify each term.

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Step 1.1.1

Simplify $2 \ln (4)$ by moving $2$ inside the logarithm.

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

Step 1.1.2

Raise $4$ to the power of $2$ .

$e^{\ln (7) + \ln (16)} - \log_{9} (\frac{\sqrt{27}}{27})$

Step 1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$e^{\ln (7 \cdot 16)} - \log_{9} (\frac{\sqrt{27}}{27})$

Step 1.3

Exponentiation and log are inverse functions.

$7 \cdot 16 - \log_{9} (\frac{\sqrt{27}}{27})$

Step 1.4

Multiply $7$ by $16$ .

$112 - \log_{9} (\frac{\sqrt{27}}{27})$

Step 1.5

Simplify the numerator.

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Step 1.5.1

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

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Step 1.5.1.1

Factor $9$ out of $27$ .

$112 - \log_{9} (\frac{\sqrt{9 (3)}}{27})$

Step 1.5.1.2

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

$112 - \log_{9} (\frac{\sqrt{3^{2} \cdot 3}}{27})$

Step 1.5.2

Pull terms out from under the radical.

$112 - \log_{9} (\frac{3 \sqrt{3}}{27})$

Step 1.6

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

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Step 1.6.1

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

$112 - \log_{9} (\frac{3 (\sqrt{3})}{27})$

Step 1.6.2

Factor $3$ out of $27$ .

$112 - \log_{9} (\frac{3 \sqrt{3}}{3 \cdot 9})$

Step 1.6.3

Cancel the common factor.

$112 - \log_{9} (\frac{3 \sqrt{3}}{3 \cdot 9})$

Step 1.6.4

Rewrite the expression.

$112 - \log_{9} (\frac{\sqrt{3}}{9})$

Step 1.7

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

$112 - \log_{9} (\sqrt{3} \cdot 9^{- 1})$

Step 1.8

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

$112 - (\log_{9} (\sqrt{3}) + \log_{9} (9^{- 1}))$

Step 1.9

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

$112 - (\log_{9} (\sqrt{3}) - \log_{9} (9))$

Step 1.10

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

$112 - (\log_{9} (\sqrt{3}) - 1 \cdot 1)$

Step 1.11

Multiply $- 1$ by $1$ .

$112 - (\log_{9} (\sqrt{3}) - 1)$

Step 1.12

Apply the distributive property.

$112 - \log_{9} (\sqrt{3}) - - 1$

Step 1.13

Multiply $- 1$ by $- 1$ .

$112 - \log_{9} (\sqrt{3}) + 1$

Step 2

Add $112$ and $1$ .

$113 - \log_{9} (\sqrt{3})$

Step 3

The result can be shown in multiple forms.

Exact Form:

$113 - \log_{9} (\sqrt{3})$

Decimal Form:

$112.75$