Solve for x 3.5=360(0.5)^(x/4)

$3.5 = 360 {(0.5)}^{\frac{x}{4}}$

Step 1

Rewrite the equation as $360 \cdot {0.5}^{\frac{x}{4}} = 3.5$ .

$360 \cdot {0.5}^{\frac{x}{4}} = 3.5$

Step 2

Divide each term in $(360 \cdot {0.5}^{\frac{x}{4}}) = 3.5$ by $360$ and simplify.

Divide each term in $(360 \cdot {0.5}^{\frac{x}{4}}) = 3.5$ by $360$ .

$\frac{(360 \cdot {0.5}^{\frac{x}{4}})}{360} = \frac{3.5}{360}$

Simplify the left side.

Cancel the common factor of $360$ .

Cancel the common factor.

$\frac{360 \cdot {0.5}^{\frac{x}{4}}}{360} = \frac{3.5}{360}$

Divide ${0.5}^{\frac{x}{4}}$ by $1$ .

${0.5}^{\frac{x}{4}} = \frac{3.5}{360}$

Simplify the right side.

Divide $3.5$ by $360$ .

${0.5}^{\frac{x}{4}} = 0.0097\overline{2}$

Step 3

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

$\ln ({0.5}^{\frac{x}{4}}) = \ln (0.0097\overline{2})$

Step 4

Expand the left side.

Expand $\ln ({0.5}^{\frac{x}{4}})$ by moving $\frac{x}{4}$ outside the logarithm.

$\frac{x}{4} \ln (0.5) = \ln (0.0097\overline{2})$

Combine $\frac{x}{4}$ and $\ln (0.5)$ .

$\frac{x \ln (0.5)}{4} = \ln (0.0097\overline{2})$

Step 5

Multiply both sides of the equation by $\frac{4}{\ln (0.5)}$ .

$\frac{4}{\ln (0.5)} \cdot \frac{x \ln (0.5)}{4} = \frac{4}{\ln (0.5)} \ln (0.0097\overline{2})$

Step 6

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{4}{\ln (0.5)} \cdot \frac{x \ln (0.5)}{4}$ .

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4}{\ln (0.5)} \cdot \frac{x \ln (0.5)}{4} = \frac{4}{\ln (0.5)} \ln (0.0097\overline{2})$

Rewrite the expression.

$\frac{1}{\ln (0.5)} (x \ln (0.5)) = \frac{4}{\ln (0.5)} \ln (0.0097\overline{2})$

Cancel the common factor of $\ln (0.5)$ .

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

$\frac{1}{\ln (0.5)} (\ln (0.5) x) = \frac{4}{\ln (0.5)} \ln (0.0097\overline{2})$

Cancel the common factor.

$\frac{1}{\ln (0.5)} (\ln (0.5) x) = \frac{4}{\ln (0.5)} \ln (0.0097\overline{2})$

Rewrite the expression.

$x = \frac{4}{\ln (0.5)} \ln (0.0097\overline{2})$

Simplify the right side.

Combine $\frac{4}{\ln (0.5)}$ and $\ln (0.0097\overline{2})$ .

$x = \frac{4 \ln (0.0097\overline{2})}{\ln (0.5)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4 \ln (0.0097\overline{2})}{\ln (0.5)}$

Decimal Form:

$x = 26.73799269 \dots$

Solve for x 3.5=360(0.5)^(x/4)

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