Precalculus Examples

$7 + \log_{2} (2 x - 1) = 5$

Step 1

Move all terms not containing $\log_{2} (2 x - 1)$ to the right side of the equation.

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Step 1.1

Subtract $7$ from both sides of the equation.

$\log_{2} (2 x - 1) = 5 - 7$

Step 1.2

Subtract $7$ from $5$ .

$\log_{2} (2 x - 1) = - 2$

Step 2

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

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

Step 3

Solve for $x$ .

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Step 3.1

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

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

Step 3.2

Simplify $2^{- 2}$ .

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Step 3.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.2.2

Raise $2$ to the power of $2$ .

$2 x - 1 = \frac{1}{4}$

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.

$2 x = \frac{1}{4} + 1$

Step 3.3.2

Write $1$ as a fraction with a common denominator.

$2 x = \frac{1}{4} + \frac{4}{4}$

Step 3.3.3

Combine the numerators over the common denominator.

$2 x = \frac{1 + 4}{4}$

Step 3.3.4

Add $1$ and $4$ .

$2 x = \frac{5}{4}$

Step 3.4

Divide each term in $2 x = \frac{5}{4}$ by $2$ and simplify.

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Step 3.4.1

Divide each term in $2 x = \frac{5}{4}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{5}{4}}{2}$

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.

$\frac{2 x}{2} = \frac{\frac{5}{4}}{2}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{5}{4}}{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5}{4} \cdot \frac{1}{2}$

Step 3.4.3.2

Multiply $\frac{5}{4} \cdot \frac{1}{2}$ .

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Step 3.4.3.2.1

Multiply $\frac{5}{4}$ by $\frac{1}{2}$ .

$x = \frac{5}{4 \cdot 2}$

Step 3.4.3.2.2

Multiply $4$ by $2$ .

$x = \frac{5}{8}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{8}$

Decimal Form:

$x = 0.625$