Algebra Examples

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

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.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $790$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

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

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

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

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

Tap for more steps...

Step 2.3.1.1

Factor $10$ out of $2500$ .

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

Step 2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.3.1.2.1

Factor $10$ out of $790$ .

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

Step 2.3.1.2.2

Cancel the common factor.

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

Step 2.3.1.2.3

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.

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

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.

Tap for more steps...

Step 7.1

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

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

Step 7.2

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.

Tap for more steps...

Step 9.1

Simplify the left side.

Tap for more steps...

Step 9.1.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 9.1.1.1

Cancel the common factor.

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

Step 9.1.1.2

Rewrite the expression.

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

Step 9.2

Simplify the right side.

Tap for more steps...

Step 9.2.1

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

Tap for more steps...

Step 9.2.1.1

Apply the distributive property.

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

Step 9.2.1.2

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.

Tap for more steps...

Step 10.1

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

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

Step 10.2

Simplify the left side.

Tap for more steps...

Step 10.2.1

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

Tap for more steps...

Step 10.2.1.1

Cancel the common factor.

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

Step 10.2.1.2

Divide $x$ by $1$ .

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

Step 10.3

Simplify the right side.

Tap for more steps...

Step 10.3.1

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

Tap for more steps...

Step 10.3.1.1

Cancel the common factor.

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

Step 10.3.1.2

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:

$x = 249.37657073 \dots$