Finite Math Examples

$\log_{6} (x) + \log_{6} (27) = 1$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{6} (x \cdot 27) = 1$

Step 1.2

Move $27$ to the left of $x$ .

$\log_{6} (27 x) = 1$

Step 2

Rewrite $\log_{6} (27 x) = 1$ 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 $b^{y} = x$ .

$6^{1} = 27 x$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $27 x = 6^{1}$ .

$27 x = 6$

Step 3.2

Divide each term in $27 x = 6^{1}$ by $27$ and simplify.

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Step 3.2.1

Divide each term in $27 x = 6^{1}$ by $27$ .

$\frac{27 x}{27} = \frac{6^{1}}{27}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $27$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{27 x}{27} = \frac{6^{1}}{27}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{6^{1}}{27}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Evaluate the exponent.

$x = \frac{6}{27}$

Step 3.2.3.2

Cancel the common factor of $6$ and $27$ .

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Step 3.2.3.2.1

Factor $3$ out of $6$ .

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

Step 3.2.3.2.2

Cancel the common factors.

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Step 3.2.3.2.2.1

Factor $3$ out of $27$ .

$x = \frac{3 \cdot 2}{3 \cdot 9}$

Step 3.2.3.2.2.2

Cancel the common factor.

$x = \frac{3 \cdot 2}{3 \cdot 9}$

Step 3.2.3.2.2.3

Rewrite the expression.

$x = \frac{2}{9}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2}{9}$

Decimal Form:

$x = 0.\overline{2}$