Algebra Examples

$\log (5 x) - \log (15.5) = 2$

Step 1

Simplify the left side.

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Step 1.1

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

$\log (\frac{5 x}{15.5}) = 2$

Step 1.2

Factor $5$ out of $5 x$ .

$\log (\frac{5 (x)}{15.5}) = 2$

Step 1.3

Factor $15.5$ out of $15.5$ .

$\log (\frac{5 (x)}{15.5 (1)}) = 2$

Step 1.4

Separate fractions.

$\log (\frac{5}{15.5} \cdot \frac{x}{1}) = 2$

Step 1.5

Divide $5$ by $15.5$ .

$\log (0.32258064 (\frac{x}{1})) = 2$

Step 1.6

Divide $x$ by $1$ .

$\log (0.32258064 x) = 2$

Step 2

Rewrite $\log (0.32258064 x) = 2$ 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 $b^{y} = x$ .

$10^{2} = 0.32258064 x$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $0.32258064 x = 10^{2}$ .

$0.32258064 x = 10^{2}$

Step 3.2

Divide each term in $0.32258064 x = 10^{2}$ by $0.32258064$ and simplify.

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Step 3.2.1

Divide each term in $0.32258064 x = 10^{2}$ by $0.32258064$ .

$\frac{0.32258064 x}{0.32258064} = \frac{10^{2}}{0.32258064}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $0.32258064$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{0.32258064 x}{0.32258064} = \frac{10^{2}}{0.32258064}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10^{2}}{0.32258064}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Raise $10$ to the power of $2$ .

$x = \frac{100}{0.32258064}$

Step 3.2.3.2

Divide $100$ by $0.32258064$ .

$x = 310$