Algebra Examples

$\log (72) - \log (\frac{2 x}{3}) = 0$

Step 1

Simplify the left side.

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Step 1.1

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

$\log (\frac{72}{\frac{2 x}{3}}) = 0$

Step 1.2

Multiply the numerator by the reciprocal of the denominator.

$\log (72 (\frac{3}{2 x})) = 0$

Step 1.3

Cancel the common factor of $2$ .

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Step 1.3.1

Factor $2$ out of $72$ .

$\log (2 (36) (\frac{3}{2 x})) = 0$

Step 1.3.2

Factor $2$ out of $2 x$ .

$\log (2 (36) (\frac{3}{2 (x)})) = 0$

Step 1.3.3

Cancel the common factor.

$\log (2 \cdot (36 (\frac{3}{2 x}))) = 0$

Step 1.3.4

Rewrite the expression.

$\log (36 (\frac{3}{x})) = 0$

Step 1.4

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

$\log (\frac{36 \cdot 3}{x}) = 0$

Step 1.5

Multiply $36$ by $3$ .

$\log (\frac{108}{x}) = 0$

Step 2

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

$10^{0} = \frac{108}{x}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $\frac{108}{x} = 10^{0}$ .

$\frac{108}{x} = 10^{0}$

Step 3.2

Anything raised to $0$ is $1$ .

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

Step 3.3

Find the LCD of the terms in the equation.

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Step 3.3.1

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

$x, 1$

Step 3.3.2

The LCM of one and any expression is the expression.

$x$

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $x$ .

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Step 3.4.2.1.1

Cancel the common factor.

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

Step 3.4.2.1.2

Rewrite the expression.

$108 = 1 x$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Multiply $x$ by $1$ .

$108 = x$

Step 3.5

Rewrite the equation as $x = 108$ .

$x = 108$