Solve for x 236=250(1.085)^(x+1)-144

$236 = 250 {(1.085)}^{x + 1} - 144$

Rewrite the equation as $250 \cdot {1.085}^{x + 1} - 144 = 236$ .

$250 \cdot {1.085}^{x + 1} - 144 = 236$

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

Add $144$ to both sides of the equation.

$250 \cdot {1.085}^{x + 1} = 236 + 144$

Add $236$ and $144$ .

$250 \cdot {1.085}^{x + 1} = 380$

Divide each term in $(250 \cdot {1.085}^{x + 1}) = 380$ by $250$ and simplify.

Divide each term in $(250 \cdot {1.085}^{x + 1}) = 380$ by $250$ .

$\frac{(250 \cdot {1.085}^{x + 1})}{250} = \frac{380}{250}$

Simplify the left side.

Cancel the common factor of $250$ .

Cancel the common factor.

$\frac{250 \cdot {1.085}^{x + 1}}{250} = \frac{380}{250}$

Divide ${1.085}^{x + 1}$ by $1$ .

${1.085}^{x + 1} = \frac{380}{250}$

Simplify the right side.

Cancel the common factor of $380$ and $250$ .

Factor $10$ out of $380$ .

${1.085}^{x + 1} = \frac{10 (38)}{250}$

Cancel the common factors.

Factor $10$ out of $250$ .

${1.085}^{x + 1} = \frac{10 \cdot 38}{10 \cdot 25}$

Cancel the common factor.

${1.085}^{x + 1} = \frac{10 \cdot 38}{10 \cdot 25}$

Rewrite the expression.

${1.085}^{x + 1} = \frac{38}{25}$

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

$\ln ({1.085}^{x + 1}) = \ln (\frac{38}{25})$

Expand $\ln ({1.085}^{x + 1})$ by moving $x + 1$ outside the logarithm.

$(x + 1) \ln (1.085) = \ln (\frac{38}{25})$

Simplify the left side.

Simplify $(x + 1) \ln (1.085)$ .

Apply the distributive property.

$x \ln (1.085) + 1 \ln (1.085) = \ln (\frac{38}{25})$

Multiply $\ln (1.085)$ by $1$ .

$x \ln (1.085) + \ln (1.085) = \ln (\frac{38}{25})$

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

$x \ln (1.085) + \ln (1.085) - \ln (\frac{38}{25}) = 0$

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x \ln (1.085) + \ln (\frac{1.085}{\frac{38}{25}}) = 0$

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$x \ln (1.085) + \ln (1.085 (\frac{25}{38})) = 0$

Multiply $1.085 (\frac{25}{38})$ .

Combine $1.085$ and $\frac{25}{38}$ .

$x \ln (1.085) + \ln (\frac{1.085 \cdot 25}{38}) = 0$

Multiply $1.085$ by $25$ .

$x \ln (1.085) + \ln (\frac{27.125}{38}) = 0$

Divide $27.125$ by $38$ .

$x \ln (1.085) + \ln (0.71381578) = 0$

Subtract $\ln (0.71381578)$ from both sides of the equation.

$x \ln (1.085) = - \ln (0.71381578)$

Divide each term in $x \ln (1.085) = - \ln (0.71381578)$ by $\ln (1.085)$ and simplify.

Divide each term in $x \ln (1.085) = - \ln (0.71381578)$ by $\ln (1.085)$ .

$\frac{x \ln (1.085)}{\ln (1.085)} = \frac{- \ln (0.71381578)}{\ln (1.085)}$

Simplify the left side.

Cancel the common factor of $\ln (1.085)$ .

Cancel the common factor.

$\frac{x \ln (1.085)}{\ln (1.085)} = \frac{- \ln (0.71381578)}{\ln (1.085)}$

Divide $x$ by $1$ .

$x = \frac{- \ln (0.71381578)}{\ln (1.085)}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{\ln (0.71381578)}{\ln (1.085)}$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (0.71381578)}{\ln (1.085)}$

Decimal Form:

$x = 4.13251289 \dots$

Solve for x 236=250(1.085)^(x+1)-144

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