Precalculus Examples

$\frac{1}{3} \cdot (2 \log_{b} (M) - \log_{b} (N) - \log_{b} (P))$

Step 1

Simplify $2 \log_{b} (M)$ by moving $2$ inside the logarithm.

$\frac{1}{3} \cdot (\log_{b} (M^{2}) - \log_{b} (N) - \log_{b} (P))$

Step 2

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

$\frac{1}{3} \cdot (\log_{b} (\frac{M^{2}}{N}) - \log_{b} (P))$

Step 3

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

$\frac{1}{3} \cdot \log_{b} (\frac{\frac{M^{2}}{N}}{P})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{3} \cdot \log_{b} (\frac{M^{2}}{N} \cdot \frac{1}{P})$

Step 5

Combine.

$\frac{1}{3} \cdot \log_{b} (\frac{M^{2} \cdot 1}{N P})$

Step 6

Multiply $M^{2}$ by $1$ .

$\frac{1}{3} \cdot \log_{b} (\frac{M^{2}}{N P})$

Step 7

Simplify $\frac{1}{3} \log_{b} (\frac{M^{2}}{N P})$ by moving $\frac{1}{3}$ inside the logarithm.

$\log_{b} ({(\frac{M^{2}}{N P})}^{\frac{1}{3}})$

Step 8

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

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Step 8.1

Apply the product rule to $\frac{M^{2}}{N P}$ .

$\log_{b} (\frac{{(M^{2})}^{\frac{1}{3}}}{{(N P)}^{\frac{1}{3}}})$

Step 8.2

Apply the product rule to $N P$ .

$\log_{b} (\frac{{(M^{2})}^{\frac{1}{3}}}{N^{\frac{1}{3}} P^{\frac{1}{3}}})$

Step 9

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

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Step 9.1

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

$\log_{b} (\frac{M^{2 (\frac{1}{3})}}{N^{\frac{1}{3}} P^{\frac{1}{3}}})$

Step 9.2

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

$\log_{b} (\frac{M^{\frac{2}{3}}}{N^{\frac{1}{3}} P^{\frac{1}{3}}})$