Precalculus Examples

$\ln (e^{3} y) + \ln (e y) - 4$

Step 1

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

$\ln (e^{3} y (e y)) - 4$

Step 2

Simplify each term.

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Step 2.1

Rewrite using the commutative property of multiplication.

$\ln (e^{3} \cdot e y \cdot y) - 4$

Step 2.2

Multiply $e^{3}$ by $e$ by adding the exponents.

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Step 2.2.1

Multiply $e^{3}$ by $e$ .

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Step 2.2.1.1

Raise $e$ to the power of $1$ .

$\ln (e^{3} \cdot e^{1} y \cdot y) - 4$

Step 2.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\ln (e^{3 + 1} y \cdot y) - 4$

Step 2.2.2

Add $3$ and $1$ .

$\ln (e^{4} y \cdot y) - 4$

Step 2.3

Multiply $y$ by $y$ by adding the exponents.

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Step 2.3.1

Move $y$ .

$\ln (e^{4} (y \cdot y)) - 4$

Step 2.3.2

Multiply $y$ by $y$ .

$\ln (e^{4} y^{2}) - 4$

Step 2.4

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

$\ln (e^{4}) + \ln (y^{2}) - 4$

Step 2.5

Use logarithm rules to move $4$ out of the exponent.

$4 \ln (e) + \ln (y^{2}) - 4$

Step 2.6

The natural logarithm of $e$ is $1$ .

$4 \cdot 1 + \ln (y^{2}) - 4$

Step 2.7

Multiply $4$ by $1$ .

$4 + \ln (y^{2}) - 4$

Step 3

Combine the opposite terms in $4 + \ln (y^{2}) - 4$ .

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Step 3.1

Subtract $4$ from $4$ .

$0 + \ln (y^{2})$

Step 3.2

Add $0$ and $\ln (y^{2})$ .

$\ln (y^{2})$