Algebra Examples

{log}_{27} (x) = \frac{2}{3}

$\log_{27} (x) = \frac{2}{3}$

Step 1

Rewrite

{log}_{27} (x) = \frac{2}{3}

$\log_{27} (x) = \frac{2}{3}$ in exponential form using the definition of a logarithm. If

x

$x$ and

b

$b$ are positive real numbers and

b \neq 1

$b \neq 1$ , then

{log}_{b} (x) = y

$\log_{b} (x) = y$ is equivalent to

b^{y} = x

$b^{y} = x$ .

27^{\frac{2}{3}} = x

$27^{\frac{2}{3}} = x$

Step 2

Solve for

x

$x$ .

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Step 2.1

Rewrite the equation as

x = 27^{\frac{2}{3}}

$x = 27^{\frac{2}{3}}$ .

x = 27^{\frac{2}{3}}

$x = 27^{\frac{2}{3}}$

Step 2.2

Simplify

27^{\frac{2}{3}}

$27^{\frac{2}{3}}$ .

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Step 2.2.1

Simplify the expression.

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Step 2.2.1.1

Rewrite

27

$27$ as

3^{3}

$3^{3}$ .

x = {(3^{3})}^{\frac{2}{3}}

$x = {(3^{3})}^{\frac{2}{3}}$

Step 2.2.1.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$x = 3^{3 (\frac{2}{3})}$

Step 2.2.2

Cancel the common factor of

$3$ .

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Step 2.2.2.1

Cancel the common factor.

$x = 3^{3 (\frac{2}{3})}$

Step 2.2.2.2

Rewrite the expression.

$x = 3^{2}$

Step 2.2.3

Raise

$3$ to the power of

$2$ .

$x = 9$