Algebra Examples

$\log_{3} (27) + \log_{3} (\frac{1}{3}) = \log_{3} (x)$

Step 1

Rewrite the equation as $\log_{3} (x) = \log_{3} (27) + \log_{3} (\frac{1}{3})$ .

$\log_{3} (x) = \log_{3} (27) + \log_{3} (\frac{1}{3})$

Step 2

Simplify the right side.

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Step 2.1

Simplify $\log_{3} (27) + \log_{3} (\frac{1}{3})$ .

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Step 2.1.1

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

$\log_{3} (x) = \log_{3} (27 (\frac{1}{3}))$

Step 2.1.2

Cancel the common factor of $3$ .

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Step 2.1.2.1

Factor $3$ out of $27$ .

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

Step 2.1.2.2

Cancel the common factor.

$\log_{3} (x) = \log_{3} (3 \cdot 9 \frac{1}{3})$

Step 2.1.2.3

Rewrite the expression.

$\log_{3} (x) = \log_{3} (9)$

Step 2.1.3

Logarithm base $3$ of $9$ is $2$ .

$\log_{3} (x) = 2$

Step 3

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

$3^{2} = x$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $x = 3^{2}$ .

$x = 3^{2}$

Step 4.2

Raise $3$ to the power of $2$ .

$x = 9$