Algebra Examples

$\log (5) - \log (8 x) = 1$

Step 1

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

$\log (\frac{5}{8 x}) = 1$

Step 2

Rewrite $\log (\frac{5}{8 x}) = 1$ 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 $b^{y} = x$ .

$10^{1} = \frac{5}{8 x}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $\frac{5}{8 x} = 10^{1}$ .

$\frac{5}{8 x} = 10$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

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

$8 x, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$8 x$

Step 3.3

Multiply each term in $\frac{5}{8 x} = 10$ by $8 x$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{5}{8 x} = 10$ by $8 x$ .

$\frac{5}{8 x} (8 x) = 10 (8 x)$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Rewrite using the commutative property of multiplication.

$8 \frac{5}{8 x} x = 10 (8 x)$

Step 3.3.2.2

Cancel the common factor of $8$ .

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Step 3.3.2.2.1

Factor $8$ out of $8 x$ .

$8 \frac{5}{8 (x)} x = 10 (8 x)$

Step 3.3.2.2.2

Cancel the common factor.

$8 \frac{5}{8 x} x = 10 (8 x)$

Step 3.3.2.2.3

Rewrite the expression.

$\frac{5}{x} x = 10 (8 x)$

Step 3.3.2.3

Cancel the common factor of $x$ .

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Step 3.3.2.3.1

Cancel the common factor.

$\frac{5}{x} x = 10 (8 x)$

Step 3.3.2.3.2

Rewrite the expression.

$5 = 10 (8 x)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Multiply $8$ by $10$ .

$5 = 80 x$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $80 x = 5$ .

$80 x = 5$

Step 3.4.2

Divide each term in $80 x = 5$ by $80$ and simplify.

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Step 3.4.2.1

Divide each term in $80 x = 5$ by $80$ .

$\frac{80 x}{80} = \frac{5}{80}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $80$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{80 x}{80} = \frac{5}{80}$

Step 3.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{80}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Cancel the common factor of $5$ and $80$ .

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Step 3.4.2.3.1.1

Factor $5$ out of $5$ .

$x = \frac{5 (1)}{80}$

Step 3.4.2.3.1.2

Cancel the common factors.

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Step 3.4.2.3.1.2.1

Factor $5$ out of $80$ .

$x = \frac{5 \cdot 1}{5 \cdot 16}$

Step 3.4.2.3.1.2.2

Cancel the common factor.

$x = \frac{5 \cdot 1}{5 \cdot 16}$

Step 3.4.2.3.1.2.3

Rewrite the expression.

$x = \frac{1}{16}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{16}$

Decimal Form:

$x = 0.0625$