Precalculus Examples

$\frac{1}{2} (\frac{1}{3} \log (x - 3) + 6 \log (x - 1)) - 10 \log (x)$

Step 1

Simplify each term.

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Step 1.1

Simplify each term.

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Step 1.1.1

Simplify $\frac{1}{3} \log (x - 3)$ by moving $\frac{1}{3}$ inside the logarithm.

$\frac{1}{2} \cdot (\log ({(x - 3)}^{\frac{1}{3}}) + 6 \log (x - 1)) - 10 \log (x)$

Step 1.1.2

Simplify $6 \log (x - 1)$ by moving $6$ inside the logarithm.

$\frac{1}{2} \cdot (\log ({(x - 3)}^{\frac{1}{3}}) + \log ({(x - 1)}^{6})) - 10 \log (x)$

Step 1.2

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

$\frac{1}{2} \cdot \log ({(x - 3)}^{\frac{1}{3}} {(x - 1)}^{6}) - 10 \log (x)$

Step 1.3

Simplify $\frac{1}{2} \log ({(x - 3)}^{\frac{1}{3}} {(x - 1)}^{6})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log ({({(x - 3)}^{\frac{1}{3}} {(x - 1)}^{6})}^{\frac{1}{2}}) - 10 \log (x)$

Step 1.4

Apply the product rule to ${(x - 3)}^{\frac{1}{3}} {(x - 1)}^{6}$ .

$\log ({({(x - 3)}^{\frac{1}{3}})}^{\frac{1}{2}} {({(x - 1)}^{6})}^{\frac{1}{2}}) - 10 \log (x)$

Step 1.5

Multiply the exponents in ${({(x - 3)}^{\frac{1}{3}})}^{\frac{1}{2}}$ .

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Step 1.5.1

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

$\log ({(x - 3)}^{\frac{1}{3} \cdot \frac{1}{2}} {({(x - 1)}^{6})}^{\frac{1}{2}}) - 10 \log (x)$

Step 1.5.2

Multiply $\frac{1}{3} \cdot \frac{1}{2}$ .

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Step 1.5.2.1

Multiply $\frac{1}{3}$ by $\frac{1}{2}$ .

$\log ({(x - 3)}^{\frac{1}{3 \cdot 2}} {({(x - 1)}^{6})}^{\frac{1}{2}}) - 10 \log (x)$

Step 1.5.2.2

Multiply $3$ by $2$ .

$\log ({(x - 3)}^{\frac{1}{6}} {({(x - 1)}^{6})}^{\frac{1}{2}}) - 10 \log (x)$

Step 1.6

Multiply the exponents in ${({(x - 1)}^{6})}^{\frac{1}{2}}$ .

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Step 1.6.1

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

$\log ({(x - 3)}^{\frac{1}{6}} {(x - 1)}^{6 (\frac{1}{2})}) - 10 \log (x)$

Step 1.6.2

Cancel the common factor of $2$ .

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Step 1.6.2.1

Factor $2$ out of $6$ .

$\log ({(x - 3)}^{\frac{1}{6}} {(x - 1)}^{2 (3) \frac{1}{2}}) - 10 \log (x)$

Step 1.6.2.2

Cancel the common factor.

$\log ({(x - 3)}^{\frac{1}{6}} {(x - 1)}^{2 \cdot 3 \frac{1}{2}}) - 10 \log (x)$

Step 1.6.2.3

Rewrite the expression.

$\log ({(x - 3)}^{\frac{1}{6}} {(x - 1)}^{3}) - 10 \log (x)$

Step 1.7

Simplify $- 10 \log (x)$ by moving $10$ inside the logarithm.

$\log ({(x - 3)}^{\frac{1}{6}} {(x - 1)}^{3}) - \log (x^{10})$

Step 2

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

$\log (\frac{{(x - 3)}^{\frac{1}{6}} {(x - 1)}^{3}}{x^{10}})$