Precalculus Examples

$\frac{1}{3} \cdot (\log_{2} (a^{5}) - \log_{2} (b c^{3}))$

Step 1

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

$\frac{1}{3} \cdot \log_{2} (\frac{a^{5}}{b c^{3}})$

Step 2

Simplify $\frac{1}{3} \log_{2} (\frac{a^{5}}{b c^{3}})$ by moving $\frac{1}{3}$ inside the logarithm.

$\log_{2} ({(\frac{a^{5}}{b c^{3}})}^{\frac{1}{3}})$

Step 3

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

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Step 3.1

Apply the product rule to $\frac{a^{5}}{b c^{3}}$ .

$\log_{2} (\frac{{(a^{5})}^{\frac{1}{3}}}{{(b c^{3})}^{\frac{1}{3}}})$

Step 3.2

Apply the product rule to $b c^{3}$ .

$\log_{2} (\frac{{(a^{5})}^{\frac{1}{3}}}{b^{\frac{1}{3}} {(c^{3})}^{\frac{1}{3}}})$

Step 4

Multiply the exponents in ${(a^{5})}^{\frac{1}{3}}$ .

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Step 4.1

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

$\log_{2} (\frac{a^{5 (\frac{1}{3})}}{b^{\frac{1}{3}} {(c^{3})}^{\frac{1}{3}}})$

Step 4.2

Combine $5$ and $\frac{1}{3}$ .

$\log_{2} (\frac{a^{\frac{5}{3}}}{b^{\frac{1}{3}} {(c^{3})}^{\frac{1}{3}}})$

Step 5

Simplify the denominator.

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Step 5.1

Multiply the exponents in ${(c^{3})}^{\frac{1}{3}}$ .

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Step 5.1.1

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

$\log_{2} (\frac{a^{\frac{5}{3}}}{b^{\frac{1}{3}} c^{3 (\frac{1}{3})}})$

Step 5.1.2

Cancel the common factor of $3$ .

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Step 5.1.2.1

Cancel the common factor.

$\log_{2} (\frac{a^{\frac{5}{3}}}{b^{\frac{1}{3}} c^{3 (\frac{1}{3})}})$

Step 5.1.2.2

Rewrite the expression.

$\log_{2} (\frac{a^{\frac{5}{3}}}{b^{\frac{1}{3}} c^{1}})$

Step 5.2

Simplify.

$\log_{2} (\frac{a^{\frac{5}{3}}}{b^{\frac{1}{3}} c})$

Step 6

Reorder factors in $\log_{2} (\frac{a^{\frac{5}{3}}}{b^{\frac{1}{3}} c})$ .

$\log_{2} (\frac{a^{\frac{5}{3}}}{c b^{\frac{1}{3}}})$