Precalculus Examples

$\frac{1}{2} \cdot (4 \ln (a) + \ln (b) - 4 \ln (c))$

Step 1

Simplify each term.

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Step 1.1

Simplify $4 \ln (a)$ by moving $4$ inside the logarithm.

$\frac{1}{2} \cdot (\ln (a^{4}) + \ln (b) - 4 \ln (c))$

Step 1.2

Simplify $- 4 \ln (c)$ by moving $4$ inside the logarithm.

$\frac{1}{2} \cdot (\ln (a^{4}) + \ln (b) - \ln (c^{4}))$

Step 2

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

$\frac{1}{2} \cdot (\ln (a^{4} b) - \ln (c^{4}))$

Step 3

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

$\frac{1}{2} \cdot \ln (\frac{a^{4} b}{c^{4}})$

Step 4

Simplify $\frac{1}{2} \ln (\frac{a^{4} b}{c^{4}})$ by moving $\frac{1}{2}$ inside the logarithm.

$\ln ({(\frac{a^{4} b}{c^{4}})}^{\frac{1}{2}})$

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1

Apply the product rule to $\frac{a^{4} b}{c^{4}}$ .

$\ln (\frac{{(a^{4} b)}^{\frac{1}{2}}}{{(c^{4})}^{\frac{1}{2}}})$

Step 5.2

Apply the product rule to $a^{4} b$ .

$\ln (\frac{{(a^{4})}^{\frac{1}{2}} b^{\frac{1}{2}}}{{(c^{4})}^{\frac{1}{2}}})$

Step 6

Multiply the exponents in ${(a^{4})}^{\frac{1}{2}}$ .

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Step 6.1

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

$\ln (\frac{a^{4 (\frac{1}{2})} b^{\frac{1}{2}}}{{(c^{4})}^{\frac{1}{2}}})$

Step 6.2

Cancel the common factor of $2$ .

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Step 6.2.1

Factor $2$ out of $4$ .

$\ln (\frac{a^{2 (2) \frac{1}{2}} b^{\frac{1}{2}}}{{(c^{4})}^{\frac{1}{2}}})$

Step 6.2.2

Cancel the common factor.

$\ln (\frac{a^{2 \cdot 2 \frac{1}{2}} b^{\frac{1}{2}}}{{(c^{4})}^{\frac{1}{2}}})$

Step 6.2.3

Rewrite the expression.

$\ln (\frac{a^{2} b^{\frac{1}{2}}}{{(c^{4})}^{\frac{1}{2}}})$

Step 7

Multiply the exponents in ${(c^{4})}^{\frac{1}{2}}$ .

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Step 7.1

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

$\ln (\frac{a^{2} b^{\frac{1}{2}}}{c^{4 (\frac{1}{2})}})$

Step 7.2

Cancel the common factor of $2$ .

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Step 7.2.1

Factor $2$ out of $4$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

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