Solve for x 2500=790(1.045)^(x/8-5)

$2500 = 790 {(1.045)}^{\frac{x}{8} - 5}$

Step 1

Rewrite the equation as $790 \cdot {1.045}^{\frac{x}{8} - 5} = 2500$ .

$790 \cdot {1.045}^{\frac{x}{8} - 5} = 2500$

Step 2

Divide each term in $(790 \cdot {1.045}^{\frac{x}{8} - 5}) = 2500$ by $790$ and simplify.

Divide each term in $(790 \cdot {1.045}^{\frac{x}{8} - 5}) = 2500$ by $790$ .

$\frac{(790 \cdot {1.045}^{\frac{x}{8} - 5})}{790} = \frac{2500}{790}$

Simplify the left side.

Cancel the common factor of $790$ .

Cancel the common factor.

$\frac{790 \cdot {1.045}^{\frac{x}{8} - 5}}{790} = \frac{2500}{790}$

Divide ${1.045}^{\frac{x}{8} - 5}$ by $1$ .

${1.045}^{\frac{x}{8} - 5} = \frac{2500}{790}$

Simplify the right side.

Cancel the common factor of $2500$ and $790$ .

Factor $10$ out of $2500$ .

${1.045}^{\frac{x}{8} - 5} = \frac{10 (250)}{790}$

Cancel the common factors.

Factor $10$ out of $790$ .

${1.045}^{\frac{x}{8} - 5} = \frac{10 \cdot 250}{10 \cdot 79}$

Cancel the common factor.

${1.045}^{\frac{x}{8} - 5} = \frac{10 \cdot 250}{10 \cdot 79}$

Rewrite the expression.

${1.045}^{\frac{x}{8} - 5} = \frac{250}{79}$

Step 3

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

$\ln ({1.045}^{\frac{x}{8} - 5}) = \ln (\frac{250}{79})$

Step 4

Expand $\ln ({1.045}^{\frac{x}{8} - 5})$ by moving $\frac{x}{8} - 5$ outside the logarithm.

$(\frac{x}{8} - 5) \ln (1.045) = \ln (\frac{250}{79})$

Step 5

Simplify the left side.

Simplify $(\frac{x}{8} - 5) \ln (1.045)$ .

Apply the distributive property.

$\frac{x}{8} \ln (1.045) - 5 \ln (1.045) = \ln (\frac{250}{79})$

Combine $\frac{x}{8}$ and $\ln (1.045)$ .

$\frac{x \ln (1.045)}{8} - 5 \ln (1.045) = \ln (\frac{250}{79})$

Step 6

Move all the terms containing a logarithm to the left side of the equation.

$\frac{x \ln (1.045)}{8} - 5 \ln (1.045) - \ln (\frac{250}{79}) = 0$

Step 7

Move all terms not containing $x$ to the right side of the equation.

Add $5 \ln (1.045)$ to both sides of the equation.

$\frac{x \ln (1.045)}{8} - \ln (\frac{250}{79}) = 5 \ln (1.045)$

Add $\ln (\frac{250}{79})$ to both sides of the equation.

$\frac{x \ln (1.045)}{8} = 5 \ln (1.045) + \ln (\frac{250}{79})$

Step 8

Multiply both sides of the equation by $8$ .

$8 \frac{x \ln (1.045)}{8} = 8 (5 \ln (1.045) + \ln (\frac{250}{79}))$

Step 9

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$8 \frac{x \ln (1.045)}{8} = 8 (5 \ln (1.045) + \ln (\frac{250}{79}))$

Rewrite the expression.

$x \ln (1.045) = 8 (5 \ln (1.045) + \ln (\frac{250}{79}))$

Simplify the right side.

Simplify $8 (5 \ln (1.045) + \ln (\frac{250}{79}))$ .

Apply the distributive property.

$x \ln (1.045) = 8 (5 \ln (1.045)) + 8 \ln (\frac{250}{79})$

Multiply $5$ by $8$ .

$x \ln (1.045) = 40 \ln (1.045) + 8 \ln (\frac{250}{79})$

Step 10

Divide each term in $x \ln (1.045) = 40 \ln (1.045) + 8 \ln (\frac{250}{79})$ by $\ln (1.045)$ and simplify.

Divide each term in $x \ln (1.045) = 40 \ln (1.045) + 8 \ln (\frac{250}{79})$ by $\ln (1.045)$ .

$\frac{x \ln (1.045)}{\ln (1.045)} = \frac{40 \ln (1.045)}{\ln (1.045)} + \frac{8 \ln (\frac{250}{79})}{\ln (1.045)}$

Simplify the left side.

Cancel the common factor of $\ln (1.045)$ .

Cancel the common factor.

$\frac{x \ln (1.045)}{\ln (1.045)} = \frac{40 \ln (1.045)}{\ln (1.045)} + \frac{8 \ln (\frac{250}{79})}{\ln (1.045)}$

Divide $x$ by $1$ .

$x = \frac{40 \ln (1.045)}{\ln (1.045)} + \frac{8 \ln (\frac{250}{79})}{\ln (1.045)}$

Simplify the right side.

Cancel the common factor of $\ln (1.045)$ .

Cancel the common factor.

$x = \frac{40 \ln (1.045)}{\ln (1.045)} + \frac{8 \ln (\frac{250}{79})}{\ln (1.045)}$

Divide $40$ by $1$ .

$x = 40 + \frac{8 \ln (\frac{250}{79})}{\ln (1.045)}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = 40 + \frac{8 \ln (\frac{250}{79})}{\ln (1.045)}$

Decimal Form:

$x = 249.37657073 \dots$

Solve for x 2500=790(1.045)^(x/8-5)

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