Precalculus Examples

$\log (x) - \log (2 x - 1) = 0$

Step 1

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

$\log (\frac{x}{2 x - 1}) = 0$

Step 2

Rewrite $\log (\frac{x}{2 x - 1}) = 0$ 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 $b^{y} = x$ .

$10^{0} = \frac{x}{2 x - 1}$

Step 3

Cross multiply to remove the fraction.

$x = 10^{0} (2 x - 1)$

Step 4

Simplify $10^{0} (2 x - 1)$ .

Tap for more steps...

Step 4.1

Anything raised to $0$ is $1$ .

$x = 1 (2 x - 1)$

Step 4.2

Multiply $2 x - 1$ by $1$ .

$x = 2 x - 1$

Step 5

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

Tap for more steps...

Step 5.1

Subtract $2 x$ from both sides of the equation.

$x - 2 x = - 1$

Step 5.2

Subtract $2 x$ from $x$ .

$- x = - 1$

Step 6

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

Tap for more steps...

Step 6.1

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

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

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Dividing two negative values results in a positive value.

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

Step 6.2.2

Divide $x$ by $1$ .

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

Step 6.3

Simplify the right side.

Tap for more steps...

Step 6.3.1

Divide $- 1$ by $- 1$ .

$x = 1$