Precalculus Examples

$\log (x) - \log (x - 3) = \ln (e) - \log (4)$

Step 1

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

$\log (\frac{x}{x - 3}) = \ln (e) - \log (4)$

Step 2

The natural logarithm of $e$ is $1$ .

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

Step 3

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

$\log (\frac{x}{x - 3}) + \log (4) = 1$

Step 4

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 5

Combine $\frac{x}{x - 3}$ and $4$ .

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

Step 6

Move $4$ to the left of $x$ .

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

Step 7

Rewrite $\log (\frac{4 x}{x - 3}) = 1$ 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 $b^{y} = x$ .

$10^{1} = \frac{4 x}{x - 3}$

Step 8

Cross multiply to remove the fraction.

$4 x = 10 (x - 3)$

Step 9

Simplify $10 (x - 3)$ .

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Step 9.1

Apply the distributive property.

$4 x = 10 x + 10 \cdot - 3$

Step 9.2

Multiply $10$ by $- 3$ .

$4 x = 10 x - 30$

Step 10

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

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Step 10.1

Subtract $10 x$ from both sides of the equation.

$4 x - 10 x = - 30$

Step 10.2

Subtract $10 x$ from $4 x$ .

$- 6 x = - 30$

Step 11

Divide each term in $- 6 x = - 30$ by $- 6$ and simplify.

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Step 11.1

Divide each term in $- 6 x = - 30$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{- 30}{- 6}$

Step 11.2

Simplify the left side.

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Step 11.2.1

Cancel the common factor of $- 6$ .

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Step 11.2.1.1

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{- 30}{- 6}$

Step 11.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 30}{- 6}$

Step 11.3

Simplify the right side.

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Step 11.3.1

Divide $- 30$ by $- 6$ .

$x = 5$