Precalculus Examples

$4 \ln (x^{5}) - 6 \ln (y^{3}) - 7 \ln (z^{7}) + \ln (x)$

Step 1

Simplify each term.

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Step 1.1

Simplify $4 \ln (x^{5})$ by moving $4$ inside the logarithm.

$\ln ({(x^{5})}^{4}) - 6 \ln (y^{3}) - 7 \ln (z^{7}) + \ln (x)$

Step 1.2

Multiply the exponents in ${(x^{5})}^{4}$ .

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Step 1.2.1

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

$\ln (x^{5 \cdot 4}) - 6 \ln (y^{3}) - 7 \ln (z^{7}) + \ln (x)$

Step 1.2.2

Multiply $5$ by $4$ .

$\ln (x^{20}) - 6 \ln (y^{3}) - 7 \ln (z^{7}) + \ln (x)$

Step 1.3

Simplify $- 6 \ln (y^{3})$ by moving $6$ inside the logarithm.

$\ln (x^{20}) - \ln ({(y^{3})}^{6}) - 7 \ln (z^{7}) + \ln (x)$

Step 1.4

Multiply the exponents in ${(y^{3})}^{6}$ .

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Step 1.4.1

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

$\ln (x^{20}) - \ln (y^{3 \cdot 6}) - 7 \ln (z^{7}) + \ln (x)$

Step 1.4.2

Multiply $3$ by $6$ .

$\ln (x^{20}) - \ln (y^{18}) - 7 \ln (z^{7}) + \ln (x)$

Step 1.5

Simplify $- 7 \ln (z^{7})$ by moving $7$ inside the logarithm.

$\ln (x^{20}) - \ln (y^{18}) - \ln ({(z^{7})}^{7}) + \ln (x)$

Step 1.6

Multiply the exponents in ${(z^{7})}^{7}$ .

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Step 1.6.1

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

$\ln (x^{20}) - \ln (y^{18}) - \ln (z^{7 \cdot 7}) + \ln (x)$

Step 1.6.2

Multiply $7$ by $7$ .

$\ln (x^{20}) - \ln (y^{18}) - \ln (z^{49}) + \ln (x)$

Step 2

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

$\ln (\frac{x^{20}}{y^{18}}) - \ln (z^{49}) + \ln (x)$

Step 3

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

$\ln (\frac{\frac{x^{20}}{y^{18}}}{z^{49}}) + \ln (x)$

Step 4

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

$\ln (\frac{\frac{x^{20}}{y^{18}}}{z^{49}} x)$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{x^{20}}{y^{18}} \cdot \frac{1}{z^{49}} x)$

Step 6

Combine.

$\ln (\frac{x^{20} \cdot 1}{y^{18} z^{49}} x)$

Step 7

Multiply $x^{20}$ by $1$ .

$\ln (\frac{x^{20}}{y^{18} z^{49}} x)$

Step 8

Multiply $\frac{x^{20}}{y^{18} z^{49}} x$ .

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Step 8.1

Combine $\frac{x^{20}}{y^{18} z^{49}}$ and $x$ .

$\ln (\frac{x^{20} x}{y^{18} z^{49}})$

Step 8.2

Multiply $x^{20}$ by $x$ by adding the exponents.

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Step 8.2.1

Multiply $x^{20}$ by $x$ .

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Step 8.2.1.1

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

$\ln (\frac{x^{20} x^{1}}{y^{18} z^{49}})$

Step 8.2.1.2

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

$\ln (\frac{x^{20 + 1}}{y^{18} z^{49}})$

Step 8.2.2

Add $20$ and $1$ .

$\ln (\frac{x^{21}}{y^{18} z^{49}})$