Precalculus Examples

$2 \ln (a) + 2 \ln (b) - \ln (a)$

Step 1

Simplify each term.

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Step 1.1

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

$\ln (a^{2}) + 2 \ln (b) - \ln (a)$

Step 1.2

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

$\ln (a^{2}) + \ln (b^{2}) - \ln (a)$

Step 2

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

$\ln (a^{2} b^{2}) - \ln (a)$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{a^{2} b^{2}}{a})$

Step 4

Cancel the common factor of $a^{2}$ and $a$ .

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Step 4.1

Factor $a$ out of $a^{2} b^{2}$ .

$\ln (\frac{a (a b^{2})}{a})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

$\ln (\frac{a (a b^{2})}{a^{1}})$

Step 4.2.2

Factor $a$ out of $a^{1}$ .

$\ln (\frac{a (a b^{2})}{a \cdot 1})$

Step 4.2.3

Cancel the common factor.

$\ln (\frac{a (a b^{2})}{a \cdot 1})$

Step 4.2.4

Rewrite the expression.

$\ln (\frac{a b^{2}}{1})$

Step 4.2.5

Divide $a b^{2}$ by $1$ .

$\ln (a b^{2})$