Algebra Examples

${(0.5)}^{\frac{x}{4.5}} = 0.1$

Step 1

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

$\ln ({0.5}^{\frac{x}{4.5}}) = \ln (0.1)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln ({0.5}^{\frac{x}{4.5}})$ by moving $\frac{x}{4.5}$ outside the logarithm.

$\frac{x}{4.5} \ln (0.5) = \ln (0.1)$

Step 2.2

Combine $\frac{x}{4.5}$ and $\ln (0.5)$ .

$\frac{x \ln (0.5)}{4.5} = \ln (0.1)$

Step 3

Simplify the left side.

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Step 3.1

Simplify $\frac{x \ln (0.5)}{4.5}$ .

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Step 3.1.1

Factor $\ln (0.5)$ out of $x \ln (0.5)$ .

$\frac{\ln (0.5) x}{4.5} = \ln (0.1)$

Step 3.1.2

Factor $4.5$ out of $4.5$ .

$\frac{\ln (0.5) x}{4.5 (1)} = \ln (0.1)$

Step 3.1.3

Separate fractions.

$\frac{\ln (0.5)}{4.5} \cdot \frac{x}{1} = \ln (0.1)$

Step 3.1.4

Replace $e$ with an approximation.

$\frac{\log_{2.71828182} (0.5)}{4.5} \cdot \frac{x}{1} = \ln (0.1)$

Step 3.1.5

Log base $2.71828182$ of $0.5$ is approximately $- 0.69314718$ .

$\frac{- 0.69314718}{4.5} \cdot \frac{x}{1} = \ln (0.1)$

Step 3.1.6

Divide $- 0.69314718$ by $4.5$ .

$- 0.1540327 \frac{x}{1} = \ln (0.1)$

Step 3.1.7

Divide $x$ by $1$ .

$- 0.1540327 x = \ln (0.1)$

Step 4

Divide each term in $- 0.1540327 x = \ln (0.1)$ by $- 0.1540327$ and simplify.

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Step 4.1

Divide each term in $- 0.1540327 x = \ln (0.1)$ by $- 0.1540327$ .

$\frac{- 0.1540327 x}{- 0.1540327} = \frac{\ln (0.1)}{- 0.1540327}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 0.1540327$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 0.1540327 x}{- 0.1540327} = \frac{\ln (0.1)}{- 0.1540327}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (0.1)}{- 0.1540327}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (0.1)}{0.1540327}$

Step 4.3.2

Replace $e$ with an approximation.

$x = - \frac{\log_{2.71828182} (0.1)}{0.1540327}$

Step 4.3.3

Log base $2.71828182$ of $0.1$ is approximately $- 2.30258509$ .

$x = - \frac{- 2.30258509}{0.1540327}$

Step 4.3.4

Divide $- 2.30258509$ by $0.1540327$ .

$x = - - 14.94867642$

Step 4.3.5

Multiply $- 1$ by $- 14.94867642$ .

$x = 14.94867642$