Algebra Examples

\log_{27} (9) = x

$\log_{27} (9) = x$

Step 1

Rewrite the equation as

x = \log_{27} (9)

$x = \log_{27} (9)$ .

x = \log_{27} (9)

$x = \log_{27} (9)$

Step 2

Logarithm base

27

$27$ of

9

$9$ is

\frac{2}{3}

$\frac{2}{3}$ .

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Step 2.1

Rewrite as an equation.

x = \log_{27} (9) = x

$x = \log_{27} (9) = x$

Step 2.2

Rewrite

\log_{27} (9) = x

$\log_{27} (9) = x$ in exponential form using the definition of a logarithm. If

x

$x$ and

b

$b$ are positive real numbers and

b

$b$ does not equal

1

$1$ , then

\log_{b} (x) = y

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

b^{y} = x

$b^{y} = x$ .

x = 27^{x} = 9

$x = 27^{x} = 9$

Step 2.3

Create expressions in the equation that all have equal bases.

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

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

Step 2.4

Rewrite

{(3^{3})}^{x}

${(3^{3})}^{x}$ as

3^{3 x}

$3^{3 x}$ .

x = 3^{3 x} = 3^{2}

$x = 3^{3 x} = 3^{2}$

Step 2.5

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

x = 3 x = 2

$x = 3 x = 2$

Step 2.6

Solve for

x

$x$ .

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

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

Step 2.7

The variable

x

$x$ is equal to

\frac{2}{3}

$\frac{2}{3}$ .

x = \frac{2}{3}

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

x = \frac{2}{3}

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

Step 3

The result can be shown in multiple forms.

Exact Form:

x = \frac{2}{3}

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

Decimal Form:

x = 0.\overline{6}

$x = 0.\overline{6}$