Algebra Examples

$2 \ln (x) + 5 \ln (2) - \ln (8)$

Step 1

Simplify each term.

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Step 1.1

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

$\ln (x^{2}) + 5 \ln (2) - \ln (8)$

Step 1.2

Simplify $5 \ln (2)$ by moving $5$ inside the logarithm.

$\ln (x^{2}) + \ln (2^{5}) - \ln (8)$

Step 1.3

Raise $2$ to the power of $5$ .

$\ln (x^{2}) + \ln (32) - \ln (8)$

Step 2

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

$\ln (x^{2} \cdot 32) - \ln (8)$

Step 3

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

$\ln (\frac{x^{2} \cdot 32}{8})$

Step 4

Cancel the common factor of $32$ and $8$ .

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Step 4.1

Factor $8$ out of $x^{2} \cdot 32$ .

$\ln (\frac{8 (x^{2} \cdot 4)}{8})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $8$ out of $8$ .

$\ln (\frac{8 (x^{2} \cdot 4)}{8 (1)})$

Step 4.2.2

Cancel the common factor.

$\ln (\frac{8 (x^{2} \cdot 4)}{8 \cdot 1})$

Step 4.2.3

Rewrite the expression.

$\ln (\frac{x^{2} \cdot 4}{1})$

Step 4.2.4

Divide $x^{2} \cdot 4$ by $1$ .

$\ln (x^{2} \cdot 4)$

Step 5

Move $4$ to the left of $x^{2}$ .

$\ln (4 x^{2})$