Precalculus Examples

$\log (x + 16) = \log (x) + \log (16)$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

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

$\log (x + 16) - \log (x) = \log (16)$

Step 1.2

Subtract $\log (16)$ from both sides of the equation.

$\log (x + 16) - \log (x) - \log (16) = 0$

Step 2

Simplify $\log (x + 16) - \log (x) - \log (16)$ .

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Step 2.1

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

$\log (\frac{x + 16}{x}) - \log (16) = 0$

Step 2.2

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

$\log (\frac{\frac{x + 16}{x}}{16}) = 0$

Step 2.3

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{x + 16}{x} \cdot \frac{1}{16}) = 0$

Step 2.4

Multiply $\frac{x + 16}{x}$ by $\frac{1}{16}$ .

$\log (\frac{x + 16}{x \cdot 16}) = 0$

Step 2.5

Move $16$ to the left of $x$ .

$\log (\frac{x + 16}{16 x}) = 0$

Step 3

Rewrite $\log (\frac{x + 16}{16 x}) = 0$ 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 $b^{y} = x$ .

$10^{0} = \frac{x + 16}{16 x}$

Step 4

Cross multiply to remove the fraction.

$x + 16 = 10^{0} (16 x)$

Step 5

Simplify $10^{0} (16 x)$ .

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Step 5.1

Remove parentheses.

$x + 16 = 10^{0} (16 x)$

Step 5.2

Anything raised to $0$ is $1$ .

$x + 16 = 1 (16 x)$

Step 5.3

Multiply $16 x$ by $1$ .

$x + 16 = 16 x$

Step 6

Move all terms containing $x$ to the left side of the equation.

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Step 6.1

Subtract $16 x$ from both sides of the equation.

$x + 16 - 16 x = 0$

Step 6.2

Subtract $16 x$ from $x$ .

$- 15 x + 16 = 0$

Step 7

Subtract $16$ from both sides of the equation.

$- 15 x = - 16$

Step 8

Divide each term in $- 15 x = - 16$ by $- 15$ and simplify.

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Step 8.1

Divide each term in $- 15 x = - 16$ by $- 15$ .

$\frac{- 15 x}{- 15} = \frac{- 16}{- 15}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $- 15$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{- 15 x}{- 15} = \frac{- 16}{- 15}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 16}{- 15}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Dividing two negative values results in a positive value.

$x = \frac{16}{15}$