Finite Math Examples

$100 = 500 {(0.12)}^{0.7 x}$

Step 1

Rewrite the equation as $500 \cdot {0.12}^{0.7 x} = 100$ .

$500 \cdot {0.12}^{0.7 x} = 100$

Step 2

Divide each term in $500 \cdot {0.12}^{0.7 x} = 100$ by $500$ and simplify.

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Step 2.1

Divide each term in $500 \cdot {0.12}^{0.7 x} = 100$ by $500$ .

$\frac{500 \cdot {0.12}^{0.7 x}}{500} = \frac{100}{500}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $500$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{500 \cdot {0.12}^{0.7 x}}{500} = \frac{100}{500}$

Step 2.2.1.2

Divide ${0.12}^{0.7 x}$ by $1$ .

${0.12}^{0.7 x} = \frac{100}{500}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $100$ and $500$ .

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Step 2.3.1.1

Factor $100$ out of $100$ .

${0.12}^{0.7 x} = \frac{100 (1)}{500}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $100$ out of $500$ .

${0.12}^{0.7 x} = \frac{100 \cdot 1}{100 \cdot 5}$

Step 2.3.1.2.2

Cancel the common factor.

${0.12}^{0.7 x} = \frac{100 \cdot 1}{100 \cdot 5}$

Step 2.3.1.2.3

Rewrite the expression.

${0.12}^{0.7 x} = \frac{1}{5}$

Step 3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({0.12}^{0.7 x}) = \ln (\frac{1}{5})$

Step 4

Expand the left side.

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Step 4.1

Expand $\ln ({0.12}^{0.7 x})$ by moving $0.7 x$ outside the logarithm.

$0.7 x \ln (0.12) = \ln (\frac{1}{5})$

Step 4.2

Multiply $\ln (0.12)$ by $0.7$ .

$- 1.48418447 x = \ln (\frac{1}{5})$

Step 5

Divide each term in $- 1.48418447 x = \ln (\frac{1}{5})$ by $- 1.48418447$ and simplify.

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Step 5.1

Divide each term in $- 1.48418447 x = \ln (\frac{1}{5})$ by $- 1.48418447$ .

$\frac{- 1.48418447 x}{- 1.48418447} = \frac{\ln (\frac{1}{5})}{- 1.48418447}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 1.48418447$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 1.48418447 x}{- 1.48418447} = \frac{\ln (\frac{1}{5})}{- 1.48418447}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{1}{5})}{- 1.48418447}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{1}{5})}{1.48418447}$

Step 5.3.2

Replace $e$ with an approximation.

$x = - \frac{\log_{2.71828182} (\frac{1}{5})}{1.48418447}$

Step 5.3.3

Divide $1$ by $5$ .

$x = - \frac{\log_{2.71828182} (0.2)}{1.48418447}$

Step 5.3.4

Log base $2.71828182$ of $0.2$ is approximately $- 1.60943791$ .

$x = - \frac{- 1.60943791}{1.48418447}$

Step 5.3.5

Divide $- 1.60943791$ by $1.48418447$ .

$x = - - 1.08439209$

Step 5.3.6

Multiply $- 1$ by $- 1.08439209$ .

$x = 1.08439209$