Precalculus Examples

$2 \ln (x^{6}) + 2 \ln (y^{3}) - 8 \ln (z^{5})$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \ln (x^{6})$ by moving $2$ inside the logarithm.

$\ln ({(x^{6})}^{2}) + 2 \ln (y^{3}) - 8 \ln (z^{5})$

Step 1.2

Multiply the exponents in ${(x^{6})}^{2}$ .

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Step 1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\ln (x^{6 \cdot 2}) + 2 \ln (y^{3}) - 8 \ln (z^{5})$

Step 1.2.2

Multiply $6$ by $2$ .

$\ln (x^{12}) + 2 \ln (y^{3}) - 8 \ln (z^{5})$

Step 1.3

Simplify $2 \ln (y^{3})$ by moving $2$ inside the logarithm.

$\ln (x^{12}) + \ln ({(y^{3})}^{2}) - 8 \ln (z^{5})$

Step 1.4

Multiply the exponents in ${(y^{3})}^{2}$ .

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Step 1.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\ln (x^{12}) + \ln (y^{3 \cdot 2}) - 8 \ln (z^{5})$

Step 1.4.2

Multiply $3$ by $2$ .

$\ln (x^{12}) + \ln (y^{6}) - 8 \ln (z^{5})$

Step 1.5

Simplify $- 8 \ln (z^{5})$ by moving $8$ inside the logarithm.

$\ln (x^{12}) + \ln (y^{6}) - \ln ({(z^{5})}^{8})$

Step 1.6

Multiply the exponents in ${(z^{5})}^{8}$ .

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Step 1.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\ln (x^{12}) + \ln (y^{6}) - \ln (z^{5 \cdot 8})$

Step 1.6.2

Multiply $5$ by $8$ .

$\ln (x^{12}) + \ln (y^{6}) - \ln (z^{40})$

Step 2

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

$\ln (x^{12} y^{6}) - \ln (z^{40})$

Step 3

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

$\ln (\frac{x^{12} y^{6}}{z^{40}})$