Solve for x 145=9430(0.895)^(x-1)+25

$145 = 9430 {(0.895)}^{x - 1} + 25$

Rewrite the equation as $9430 \cdot {0.895}^{x - 1} + 25 = 145$ .

$9430 \cdot {0.895}^{x - 1} + 25 = 145$

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

Subtract $25$ from both sides of the equation.

$9430 \cdot {0.895}^{x - 1} = 145 - 25$

Subtract $25$ from $145$ .

$9430 \cdot {0.895}^{x - 1} = 120$

Divide each term in $(9430 \cdot {0.895}^{x - 1}) = 120$ by $9430$ and simplify.

Divide each term in $(9430 \cdot {0.895}^{x - 1}) = 120$ by $9430$ .

$\frac{(9430 \cdot {0.895}^{x - 1})}{9430} = \frac{120}{9430}$

Simplify the left side.

Cancel the common factor of $9430$ .

Cancel the common factor.

$\frac{9430 \cdot {0.895}^{x - 1}}{9430} = \frac{120}{9430}$

Divide ${0.895}^{x - 1}$ by $1$ .

${0.895}^{x - 1} = \frac{120}{9430}$

Simplify the right side.

Cancel the common factor of $120$ and $9430$ .

Factor $10$ out of $120$ .

${0.895}^{x - 1} = \frac{10 (12)}{9430}$

Cancel the common factors.

Factor $10$ out of $9430$ .

${0.895}^{x - 1} = \frac{10 \cdot 12}{10 \cdot 943}$

Cancel the common factor.

${0.895}^{x - 1} = \frac{10 \cdot 12}{10 \cdot 943}$

Rewrite the expression.

${0.895}^{x - 1} = \frac{12}{943}$

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

$\ln ({0.895}^{x - 1}) = \ln (\frac{12}{943})$

Expand $\ln ({0.895}^{x - 1})$ by moving $x - 1$ outside the logarithm.

$(x - 1) \ln (0.895) = \ln (\frac{12}{943})$

Simplify the left side.

Simplify $(x - 1) \ln (0.895)$ .

Apply the distributive property.

$x \ln (0.895) - 1 \ln (0.895) = \ln (\frac{12}{943})$

Rewrite $- 1 \ln (0.895)$ as $- \ln (0.895)$ .

$x \ln (0.895) - \ln (0.895) = \ln (\frac{12}{943})$

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

$x \ln (0.895) - \ln (0.895) - \ln (\frac{12}{943}) = 0$

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

Add $\ln (0.895)$ to both sides of the equation.

$x \ln (0.895) - \ln (\frac{12}{943}) = \ln (0.895)$

Add $\ln (\frac{12}{943})$ to both sides of the equation.

$x \ln (0.895) = \ln (0.895) + \ln (\frac{12}{943})$

Divide each term in $x \ln (0.895) = \ln (0.895) + \ln (\frac{12}{943})$ by $\ln (0.895)$ and simplify.

Divide each term in $x \ln (0.895) = \ln (0.895) + \ln (\frac{12}{943})$ by $\ln (0.895)$ .

$\frac{x \ln (0.895)}{\ln (0.895)} = \frac{\ln (0.895)}{\ln (0.895)} + \frac{\ln (\frac{12}{943})}{\ln (0.895)}$

Simplify the left side.

Cancel the common factor of $\ln (0.895)$ .

Cancel the common factor.

$\frac{x \ln (0.895)}{\ln (0.895)} = \frac{\ln (0.895)}{\ln (0.895)} + \frac{\ln (\frac{12}{943})}{\ln (0.895)}$

Divide $x$ by $1$ .

$x = \frac{\ln (0.895)}{\ln (0.895)} + \frac{\ln (\frac{12}{943})}{\ln (0.895)}$

Simplify the right side.

Cancel the common factor of $\ln (0.895)$ .

Cancel the common factor.

$x = \frac{\ln (0.895)}{\ln (0.895)} + \frac{\ln (\frac{12}{943})}{\ln (0.895)}$

Rewrite the expression.

$x = 1 + \frac{\ln (\frac{12}{943})}{\ln (0.895)}$

The result can be shown in multiple forms.

Exact Form:

$x = 1 + \frac{\ln (\frac{12}{943})}{\ln (0.895)}$

Decimal Form:

$x = 40.34100994 \dots$

Solve for x 145=9430(0.895)^(x-1)+25

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