Precalculus Examples

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

Step 1

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (2 x - 1)} = e^{0}$

Step 2

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

$e^{0} = 2 x - 1$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $2 x - 1 = e^{0}$ .

$2 x - 1 = e^{0}$

Step 3.2

Anything raised to $0$ is $1$ .

$2 x - 1 = 1$

Step 3.3

Move all terms not containing $x$ to the right side of the equation.

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Step 3.3.1

Add $1$ to both sides of the equation.

$2 x = 1 + 1$

Step 3.3.2

Add $1$ and $1$ .

$2 x = 2$

Step 3.4

Divide each term in $2 x = 2$ by $2$ and simplify.

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Step 3.4.1

Divide each term in $2 x = 2$ by $2$ .

$\frac{2 x}{2} = \frac{2}{2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{2}{2}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $2$ by $2$ .

$x = 1$