Precalculus Examples

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

Step 1

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

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

Step 2

Logarithm base $2$ of $4$ is $2$ .

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

Step 3

Simplify the left side.

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Step 3.1

Simplify $3 \log_{2} (x - 1)$ by moving $3$ inside the logarithm.

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

Step 4

Move all terms not containing $\log_{2} ({(x - 1)}^{3})$ to the right side of the equation.

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Step 4.1

Subtract $2$ from both sides of the equation.

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

Step 4.2

Subtract $2$ from $5$ .

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

Step 5

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

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

Step 6

Solve for $x$ .

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Step 6.1

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

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

Step 6.2

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$x - 1 = 2$

Step 6.3

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

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Step 6.3.1

Add $1$ to both sides of the equation.

$x = 2 + 1$

Step 6.3.2

Add $2$ and $1$ .

$x = 3$