Precalculus Examples

$\log (P) = \log (c) - k \log (W)$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$\log (P) - \log (c) + k \log (W) = 0$

Step 2

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

$\log (\frac{P}{c}) + k \log (W) = 0$

Step 3

Subtract $k \log (W)$ from both sides of the equation.

$\log (\frac{P}{c}) = - k \log (W)$

Step 4

Rewrite $\log (\frac{P}{c}) = - k \log (W)$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{- k \log (W)} = \frac{P}{c}$

Step 5

Solve for $P$ .

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Step 5.1

Rewrite the equation as $\frac{P}{c} = 10^{- k \log (W)}$ .

$\frac{P}{c} = 10^{- k \log (W)}$

Step 5.2

Multiply both sides by $c$ .

$\frac{P}{c} c = 10^{- k \log (W)} c$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Cancel the common factor of $c$ .

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Step 5.3.1.1.1

Cancel the common factor.

$\frac{P}{c} c = 10^{- k \log (W)} c$

Step 5.3.1.1.2

Rewrite the expression.

$P = 10^{- k \log (W)} c$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Reorder factors in $10^{- k \log (W)} c$ .

$P = c \cdot 10^{- k \log (W)}$