Precalculus Examples

$\log (4 x) = \log (5) + \log (x - 3)$

Step 1

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

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Step 1.1

Subtract $\log (5)$ from both sides of the equation.

$\log (4 x) - \log (5) = \log (x - 3)$

Step 1.2

Subtract $\log (x - 3)$ from both sides of the equation.

$\log (4 x) - \log (5) - \log (x - 3) = 0$

Step 2

Simplify $\log (4 x) - \log (5) - \log (x - 3)$ .

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Step 2.1

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

$\log (\frac{4 x}{5}) - \log (x - 3) = 0$

Step 2.2

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

$\log (\frac{\frac{4 x}{5}}{x - 3}) = 0$

Step 2.3

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{4 x}{5} \cdot \frac{1}{x - 3}) = 0$

Step 2.4

Multiply $\frac{4 x}{5}$ by $\frac{1}{x - 3}$ .

$\log (\frac{4 x}{5 (x - 3)}) = 0$

Step 3

Rewrite $\log (\frac{4 x}{5 (x - 3)}) = 0$ 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 $b^{y} = x$ .

$10^{0} = \frac{4 x}{5 (x - 3)}$

Step 4

Cross multiply to remove the fraction.

$4 x = 10^{0} (5 (x - 3))$

Step 5

Simplify $10^{0} (5 (x - 3))$ .

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Step 5.1

Simplify the expression.

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Step 5.1.1

Anything raised to $0$ is $1$ .

$4 x = 1 (5 (x - 3))$

Step 5.1.2

Multiply $5 (x - 3)$ by $1$ .

$4 x = 5 (x - 3)$

Step 5.2

Apply the distributive property.

$4 x = 5 x + 5 \cdot - 3$

Step 5.3

Multiply $5$ by $- 3$ .

$4 x = 5 x - 15$

Step 6

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

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Step 6.1

Subtract $5 x$ from both sides of the equation.

$4 x - 5 x = - 15$

Step 6.2

Subtract $5 x$ from $4 x$ .

$- x = - 15$

Step 7

Divide each term in $- x = - 15$ by $- 1$ and simplify.

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Step 7.1

Divide each term in $- x = - 15$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 15}{- 1}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 15}{- 1}$

Step 7.2.2

Divide $x$ by $1$ .

$x = \frac{- 15}{- 1}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $- 15$ by $- 1$ .

$x = 15$