Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Multiply $4$ by $4$ .

$\ln (x^{16}) - 5 \ln (y^{3}) - 3 \ln (x^{7}) + \ln (x)$

Step 1.3

Simplify $- 5 \ln (y^{3})$ by moving $5$ inside the logarithm.

$\ln (x^{16}) - \ln ({(y^{3})}^{5}) - 3 \ln (x^{7}) + \ln (x)$

Step 1.4

Multiply the exponents in ${(y^{3})}^{5}$ .

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Step 1.4.1

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

$\ln (x^{16}) - \ln (y^{3 \cdot 5}) - 3 \ln (x^{7}) + \ln (x)$

Step 1.4.2

Multiply $3$ by $5$ .

$\ln (x^{16}) - \ln (y^{15}) - 3 \ln (x^{7}) + \ln (x)$

Step 1.5

Simplify $- 3 \ln (x^{7})$ by moving $3$ inside the logarithm.

$\ln (x^{16}) - \ln (y^{15}) - \ln ({(x^{7})}^{3}) + \ln (x)$

Step 1.6

Multiply the exponents in ${(x^{7})}^{3}$ .

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Step 1.6.1

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

$\ln (x^{16}) - \ln (y^{15}) - \ln (x^{7 \cdot 3}) + \ln (x)$

Step 1.6.2

Multiply $7$ by $3$ .

$\ln (x^{16}) - \ln (y^{15}) - \ln (x^{21}) + \ln (x)$

Step 2

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

$\ln (\frac{x^{16}}{y^{15}}) - \ln (x^{21}) + \ln (x)$

Step 3

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

$\ln (\frac{\frac{x^{16}}{y^{15}}}{x^{21}}) + \ln (x)$

Step 4

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

$\ln (\frac{\frac{x^{16}}{y^{15}}}{x^{21}} x)$

Step 5

Cancel the common factor of $x$ .

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Step 5.1

Factor $x$ out of $x^{21}$ .

$\ln (\frac{\frac{x^{16}}{y^{15}}}{x \cdot x^{20}} x)$

Step 5.2

Cancel the common factor.

$\ln (\frac{\frac{x^{16}}{y^{15}}}{x \cdot x^{20}} x)$

Step 5.3

Rewrite the expression.

$\ln (\frac{\frac{x^{16}}{y^{15}}}{x^{20}})$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{x^{16}}{y^{15}} \cdot \frac{1}{x^{20}})$

Step 7

Combine.

$\ln (\frac{x^{16} \cdot 1}{y^{15} x^{20}})$

Step 8

Cancel the common factor of $x^{16}$ and $x^{20}$ .

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Step 8.1

Factor $x^{16}$ out of $x^{16} \cdot 1$ .

$\ln (\frac{x^{16} (1)}{y^{15} x^{20}})$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $x^{16}$ out of $y^{15} x^{20}$ .

$\ln (\frac{x^{16} (1)}{x^{16} (y^{15} x^{4})})$

Step 8.2.2

Cancel the common factor.

$\ln (\frac{x^{16} \cdot 1}{x^{16} (y^{15} x^{4})})$

Step 8.2.3

Rewrite the expression.

$\ln (\frac{1}{y^{15} x^{4}})$