Algebra Examples

$\log_{8} (48) - \log_{8} (x) = \log_{8} (4)$

Step 1

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

$\log_{8} (\frac{48}{x}) = \log_{8} (4)$

Step 2

Logarithm base $8$ of $4$ is $\frac{2}{3}$ .

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Step 2.1

Rewrite as an equation.

$\log_{8} (4) = x$

Step 2.2

Rewrite $\log_{8} (4) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ does not equal $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$8^{x} = 4$

Step 2.3

Create expressions in the equation that all have equal bases.

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

Step 2.4

Rewrite ${(2^{3})}^{x}$ as $2^{3 x}$ .

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

Step 2.5

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

$3 x = 2$

Step 2.6

Solve for $x$ .

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

Step 2.7

The variable $x$ is equal to $\frac{2}{3}$ .

$\log_{8} (\frac{48}{x}) = \frac{2}{3}$

Step 3

Rewrite $\log_{8} (\frac{48}{x}) = \frac{2}{3}$ 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 $b^{y} = x$ .

$8^{\frac{2}{3}} = \frac{48}{x}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $\frac{48}{x} = 8^{\frac{2}{3}}$ .

$\frac{48}{x} = 8^{\frac{2}{3}}$

Step 4.2

Factor each term.

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Step 4.2.1

Rewrite $8$ as $2^{3}$ .

$\frac{48}{x} = {(2^{3})}^{\frac{2}{3}}$

Step 4.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{48}{x} = 2^{3 (\frac{2}{3})}$

Step 4.2.3

Cancel the common factor of $3$ .

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Step 4.2.3.1

Cancel the common factor.

$\frac{48}{x} = 2^{3 (\frac{2}{3})}$

Step 4.2.3.2

Rewrite the expression.

$\frac{48}{x} = 2^{2}$

Step 4.2.4

Raise $2$ to the power of $2$ .

$\frac{48}{x} = 4$

Step 4.3

Find the LCD of the terms in the equation.

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Step 4.3.1

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

$x, 1$

Step 4.3.2

The LCM of one and any expression is the expression.

$x$

Step 4.4

Multiply each term in $\frac{48}{x} = 4$ by $x$ to eliminate the fractions.

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Step 4.4.1

Multiply each term in $\frac{48}{x} = 4$ by $x$ .

$\frac{48}{x} x = 4 x$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $x$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{48}{x} x = 4 x$

Step 4.4.2.1.2

Rewrite the expression.

$48 = 4 x$

Step 4.5

Solve the equation.

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Step 4.5.1

Rewrite the equation as $4 x = 48$ .

$4 x = 48$

Step 4.5.2

Divide each term in $4 x = 48$ by $4$ and simplify.

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Step 4.5.2.1

Divide each term in $4 x = 48$ by $4$ .

$\frac{4 x}{4} = \frac{48}{4}$

Step 4.5.2.2

Simplify the left side.

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Step 4.5.2.2.1

Cancel the common factor of $4$ .

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Step 4.5.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{48}{4}$

Step 4.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{48}{4}$

Step 4.5.2.3

Simplify the right side.

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Step 4.5.2.3.1

Divide $48$ by $4$ .

$x = 12$