Precalculus Examples

$\frac{1}{3} \cdot \log (27 x^{2}) - 3 \log (y) + \log (z)$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{3} \log (27 x^{2})$ by moving $\frac{1}{3}$ inside the logarithm.

$\log ({(27 x^{2})}^{\frac{1}{3}}) - 3 \log (y) + \log (z)$

Step 1.2

Apply the product rule to $27 x^{2}$ .

$\log (27^{\frac{1}{3}} {(x^{2})}^{\frac{1}{3}}) - 3 \log (y) + \log (z)$

Step 1.3

Rewrite $27$ as $3^{3}$ .

$\log ({(3^{3})}^{\frac{1}{3}} {(x^{2})}^{\frac{1}{3}}) - 3 \log (y) + \log (z)$

Step 1.4

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

$\log (3^{3 (\frac{1}{3})} {(x^{2})}^{\frac{1}{3}}) - 3 \log (y) + \log (z)$

Step 1.5

Cancel the common factor of $3$ .

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Step 1.5.1

Cancel the common factor.

$\log (3^{3 (\frac{1}{3})} {(x^{2})}^{\frac{1}{3}}) - 3 \log (y) + \log (z)$

Step 1.5.2

Rewrite the expression.

$\log (3^{1} {(x^{2})}^{\frac{1}{3}}) - 3 \log (y) + \log (z)$

Step 1.6

Evaluate the exponent.

$\log (3 {(x^{2})}^{\frac{1}{3}}) - 3 \log (y) + \log (z)$

Step 1.7

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

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Step 1.7.1

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

$\log (3 x^{2 (\frac{1}{3})}) - 3 \log (y) + \log (z)$

Step 1.7.2

Combine $2$ and $\frac{1}{3}$ .

$\log (3 x^{\frac{2}{3}}) - 3 \log (y) + \log (z)$

Step 1.8

Simplify $- 3 \log (y)$ by moving $3$ inside the logarithm.

$\log (3 x^{\frac{2}{3}}) - \log (y^{3}) + \log (z)$

Step 2

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

$\log (\frac{3 x^{\frac{2}{3}}}{y^{3}}) + \log (z)$

Step 3

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

$\log (\frac{3 x^{\frac{2}{3}}}{y^{3}} z)$

Step 4

Combine $\frac{3 x^{\frac{2}{3}}}{y^{3}}$ and $z$ .

$\log (\frac{3 x^{\frac{2}{3}} z}{y^{3}})$

Step 5

Reorder factors in $\log (\frac{3 x^{\frac{2}{3}} z}{y^{3}})$ .

$\log (\frac{3 z x^{\frac{2}{3}}}{y^{3}})$