Finite Math Examples

$108 = 10 \log (\frac{10 s}{10^{- 12}})$

Step 1

Rewrite the equation as $10 \log (\frac{10 s}{10^{- 12}}) = 108$ .

$10 \log (\frac{10 s}{10^{- 12}}) = 108$

Step 2

Simplify.

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Step 2.1

Move $10^{- 12}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$10 \log (10 s \cdot 10^{12}) = 108$

Step 2.2

Multiply $10$ by $10^{12}$ by adding the exponents.

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Step 2.2.1

Move $10^{12}$ .

$10 \log (10^{12} \cdot 10 s) = 108$

Step 2.2.2

Multiply $10^{12}$ by $10$ .

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Step 2.2.2.1

Raise $10$ to the power of $1$ .

$10 \log (10^{12} \cdot 10^{1} s) = 108$

Step 2.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$10 \log (10^{12 + 1} s) = 108$

Step 2.2.3

Add $12$ and $1$ .

$10 \log (10^{13} s) = 108$

Step 3

Divide each term in $10 \log (10^{13} s) = 108$ by $10$ and simplify.

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Step 3.1

Divide each term in $10 \log (10^{13} s) = 108$ by $10$ .

$\frac{10 \log (10^{13} s)}{10} = \frac{108}{10}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $10$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{10 \log (10^{13} s)}{10} = \frac{108}{10}$

Step 3.2.1.2

Divide $\log (10^{13} s)$ by $1$ .

$\log (10^{13} s) = \frac{108}{10}$

Step 3.2.2

Rewrite $\log (10^{13} s)$ as $\log (10^{13}) + \log (s)$ .

$\log (10^{13}) + \log (s) = \frac{108}{10}$

Step 3.2.3

Use logarithm rules to move $13$ out of the exponent.

$13 \log (10) + \log (s) = \frac{108}{10}$

Step 3.2.4

Logarithm base $10$ of $10$ is $1$ .

$13 \cdot 1 + \log (s) = \frac{108}{10}$

Step 3.2.5

Multiply $13$ by $1$ .

$13 + \log (s) = \frac{108}{10}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $108$ and $10$ .

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Step 3.3.1.1

Factor $2$ out of $108$ .

$13 + \log (s) = \frac{2 (54)}{10}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $2$ out of $10$ .

$13 + \log (s) = \frac{2 \cdot 54}{2 \cdot 5}$

Step 3.3.1.2.2

Cancel the common factor.

$13 + \log (s) = \frac{2 \cdot 54}{2 \cdot 5}$

Step 3.3.1.2.3

Rewrite the expression.

$13 + \log (s) = \frac{54}{5}$

Step 4

Move all terms not containing $\log (s)$ to the right side of the equation.

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Step 4.1

Subtract $13$ from both sides of the equation.

$\log (s) = \frac{54}{5} - 13$

Step 4.2

To write $- 13$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\log (s) = \frac{54}{5} - 13 \cdot \frac{5}{5}$

Step 4.3

Combine $- 13$ and $\frac{5}{5}$ .

$\log (s) = \frac{54}{5} + \frac{- 13 \cdot 5}{5}$

Step 4.4

Combine the numerators over the common denominator.

$\log (s) = \frac{54 - 13 \cdot 5}{5}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 13$ by $5$ .

$\log (s) = \frac{54 - 65}{5}$

Step 4.5.2

Subtract $65$ from $54$ .

$\log (s) = \frac{- 11}{5}$

Step 4.6

Move the negative in front of the fraction.

$\log (s) = - \frac{11}{5}$

Step 5

Rewrite $\log (s) = - \frac{11}{5}$ 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 $b^{y} = x$ .

$10^{- \frac{11}{5}} = s$

Step 6

Solve for $s$ .

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Step 6.1

Rewrite the equation as $s = 10^{- \frac{11}{5}}$ .

$s = 10^{- \frac{11}{5}}$

Step 6.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$s = \frac{1}{10^{\frac{11}{5}}}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$s = \frac{1}{10^{\frac{11}{5}}}$

Decimal Form:

$s = 0.00630957 \dots$