Precalculus Examples

$2 \log_{2} (x - 9) + \log_{2} (2) = 3$

Step 1

Logarithm base $2$ of $2$ is $1$ .

$2 \log_{2} (x - 9) + 1 = 3$

Step 2

Move all terms not containing $\log_{2} (x - 9)$ to the right side of the equation.

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Step 2.1

Subtract $1$ from both sides of the equation.

$2 \log_{2} (x - 9) = 3 - 1$

Step 2.2

Subtract $1$ from $3$ .

$2 \log_{2} (x - 9) = 2$

Step 3

Divide each term in $2 \log_{2} (x - 9) = 2$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 \log_{2} (x - 9) = 2$ by $2$ .

$\frac{2 \log_{2} (x - 9)}{2} = \frac{2}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 \log_{2} (x - 9)}{2} = \frac{2}{2}$

Step 3.2.1.2

Divide $\log_{2} (x - 9)$ by $1$ .

$\log_{2} (x - 9) = \frac{2}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $2$ by $2$ .

$\log_{2} (x - 9) = 1$

Step 4

Rewrite $\log_{2} (x - 9) = 1$ 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 $b^{y} = x$ .

$2^{1} = x - 9$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $x - 9 = 2^{1}$ .

$x - 9 = 2$

Step 5.2

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

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Step 5.2.1

Add $9$ to both sides of the equation.

$x = 2 + 9$

Step 5.2.2

Add $2$ and $9$ .

$x = 11$