Precalculus Examples

$\log_{2} ({(x - 1)}^{3}) + \log_{2} (4) = 5$

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

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

$\log_{2} ({(x - 1)}^{3} \cdot 4) = 5$

Step 1.2

Move $4$ to the left of ${(x - 1)}^{3}$ .

$\log_{2} (4 {(x - 1)}^{3}) = 5$

Step 2

Rewrite $\log_{2} (4 {(x - 1)}^{3}) = 5$ 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 $b^{y} = x$ .

$2^{5} = 4 {(x - 1)}^{3}$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $4 {(x - 1)}^{3} = 2^{5}$ .

$4 {(x - 1)}^{3} = 2^{5}$

Step 3.2

Divide each term in $4 {(x - 1)}^{3} = 2^{5}$ by $4$ and simplify.

Tap for more steps...

Step 3.2.1

Divide each term in $4 {(x - 1)}^{3} = 2^{5}$ by $4$ .

$\frac{4 {(x - 1)}^{3}}{4} = \frac{2^{5}}{4}$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 {(x - 1)}^{3}}{4} = \frac{2^{5}}{4}$

Step 3.2.2.1.2

Divide ${(x - 1)}^{3}$ by $1$ .

${(x - 1)}^{3} = \frac{2^{5}}{4}$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Raise $2$ to the power of $5$ .

${(x - 1)}^{3} = \frac{32}{4}$

Step 3.2.3.2

Divide $32$ by $4$ .

${(x - 1)}^{3} = 8$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x - 1 = \sqrt[3]{8}$

Step 3.4

Simplify $\sqrt[3]{8}$ .

Tap for more steps...

Step 3.4.1

Rewrite $8$ as $2^{3}$ .

$x - 1 = \sqrt[3]{2^{3}}$

Step 3.4.2

Pull terms out from under the radical, assuming real numbers.

$x - 1 = 2$

Step 3.5

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

Tap for more steps...

Step 3.5.1

Add $1$ to both sides of the equation.

$x = 2 + 1$

Step 3.5.2

Add $2$ and $1$ .

$x = 3$