Finite Math Examples

$\frac{1}{3} \cdot \log_{2} (27) + \log_{2} (36) = \log_{2} (g)$

Step 1

Simplify the left side.

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Step 1.1

Combine $\frac{1}{3}$ and $\log_{2} (27)$ .

$\frac{\log_{2} (27)}{3} + \log_{2} (36) = \log_{2} (g)$

Step 2

Move all the terms containing a logarithm to the left side of the equation.

$\frac{\log_{2} (27)}{3} + \log_{2} (36) - \log_{2} (g) = 0$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{\log_{2} (27)}{3} + \log_{2} (\frac{36}{g}) = 0$

Step 4

Simplify each term.

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Step 4.1

Rewrite $\log_{2} (27)$ as $\log_{2} (3^{3})$ .

$\frac{\log_{2} (3^{3})}{3} + \log_{2} (\frac{36}{g}) = 0$

Step 4.2

Expand $\log_{2} (3^{3})$ by moving $3$ outside the logarithm.

$\frac{3 \log_{2} (3)}{3} + \log_{2} (\frac{36}{g}) = 0$

Step 4.3

Cancel the common factor of $3$ .

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Step 4.3.1

Cancel the common factor.

$\frac{3 \log_{2} (3)}{3} + \log_{2} (\frac{36}{g}) = 0$

Step 4.3.2

Divide $\log_{2} (3)$ by $1$ .

$\log_{2} (3) + \log_{2} (\frac{36}{g}) = 0$

Step 5

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

$\log_{2} (3 (\frac{36}{g})) = 0$

Step 6

Multiply $3 \frac{36}{g}$ .

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Step 6.1

Combine $3$ and $\frac{36}{g}$ .

$\log_{2} (\frac{3 \cdot 36}{g}) = 0$

Step 6.2

Multiply $3$ by $36$ .

$\log_{2} (\frac{108}{g}) = 0$

Step 7

Rewrite $\log_{2} (\frac{108}{g}) = 0$ 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 $b^{y} = x$ .

$2^{0} = \frac{108}{g}$

Step 8

Solve for $g$ .

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Step 8.1

Rewrite the equation as $\frac{108}{g} = 2^{0}$ .

$\frac{108}{g} = 2^{0}$

Step 8.2

Anything raised to $0$ is $1$ .

$\frac{108}{g} = 1$

Step 8.3

Find the LCD of the terms in the equation.

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Step 8.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$g, 1$

Step 8.3.2

The LCM of one and any expression is the expression.

$g$

Step 8.4

Multiply each term in $\frac{108}{g} = 1$ by $g$ to eliminate the fractions.

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Step 8.4.1

Multiply each term in $\frac{108}{g} = 1$ by $g$ .

$\frac{108}{g} g = 1 g$

Step 8.4.2

Simplify the left side.

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Step 8.4.2.1

Cancel the common factor of $g$ .

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Step 8.4.2.1.1

Cancel the common factor.

$\frac{108}{g} g = 1 g$

Step 8.4.2.1.2

Rewrite the expression.

$108 = 1 g$

Step 8.4.3

Simplify the right side.

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Step 8.4.3.1

Multiply $g$ by $1$ .

$108 = g$

Step 8.5

Rewrite the equation as $g = 108$ .

$g = 108$