Precalculus Examples

$6 \ln (P Q) - \ln (\frac{P}{Q})$

Step 1

Simplify each term.

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Step 1.1

Simplify $6 \ln (P Q)$ by moving $6$ inside the logarithm.

$\ln ({(P Q)}^{6}) - \ln (\frac{P}{Q})$

Step 1.2

Apply the product rule to $P Q$ .

$\ln (P^{6} Q^{6}) - \ln (\frac{P}{Q})$

Step 2

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

$\ln (\frac{P^{6} Q^{6}}{\frac{P}{Q}})$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\ln (P^{6} Q^{6} \frac{Q}{P})$

Step 4

Cancel the common factor of $P$ .

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Step 4.1

Factor $P$ out of $P^{6} Q^{6}$ .

$\ln (P (P^{5} Q^{6}) \frac{Q}{P})$

Step 4.2

Cancel the common factor.

$\ln (P (P^{5} Q^{6}) \frac{Q}{P})$

Step 4.3

Rewrite the expression.

$\ln (P^{5} Q^{6} Q)$

Step 5

Raise $Q$ to the power of $1$ .

$\ln (P^{5} (Q^{1} Q^{6}))$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\ln (P^{5} Q^{1 + 6})$

Step 7

Add $1$ and $6$ .

$\ln (P^{5} Q^{7})$