Solve for x 1410=4500(0.95)^(2x)+580

$1410 = 4500 {(0.95)}^{2 x} + 580$

Rewrite the equation as $4500 \cdot {0.95}^{2 x} + 580 = 1410$ .

$4500 \cdot {0.95}^{2 x} + 580 = 1410$

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

Subtract $580$ from both sides of the equation.

$4500 \cdot {0.95}^{2 x} = 1410 - 580$

Subtract $580$ from $1410$ .

$4500 \cdot {0.95}^{2 x} = 830$

Divide each term in $(4500 \cdot {0.95}^{2 x}) = 830$ by $4500$ and simplify.

Divide each term in $(4500 \cdot {0.95}^{2 x}) = 830$ by $4500$ .

$\frac{(4500 \cdot {0.95}^{2 x})}{4500} = \frac{830}{4500}$

Simplify the left side.

Cancel the common factor of $4500$ .

Cancel the common factor.

$\frac{4500 \cdot {0.95}^{2 x}}{4500} = \frac{830}{4500}$

Divide ${0.95}^{2 x}$ by $1$ .

${0.95}^{2 x} = \frac{830}{4500}$

Simplify the right side.

Cancel the common factor of $830$ and $4500$ .

Factor $10$ out of $830$ .

${0.95}^{2 x} = \frac{10 (83)}{4500}$

Cancel the common factors.

Factor $10$ out of $4500$ .

${0.95}^{2 x} = \frac{10 \cdot 83}{10 \cdot 450}$

Cancel the common factor.

${0.95}^{2 x} = \frac{10 \cdot 83}{10 \cdot 450}$

Rewrite the expression.

${0.95}^{2 x} = \frac{83}{450}$

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

$\ln ({0.95}^{2 x}) = \ln (\frac{83}{450})$

Expand $\ln ({0.95}^{2 x})$ by moving $2 x$ outside the logarithm.

$2 x \ln (0.95) = \ln (\frac{83}{450})$

Divide each term in $2 x \ln (0.95) = \ln (\frac{83}{450})$ by $2 \ln (0.95)$ and simplify.

Divide each term in $2 x \ln (0.95) = \ln (\frac{83}{450})$ by $2 \ln (0.95)$ .

$\frac{2 x \ln (0.95)}{2 \ln (0.95)} = \frac{\ln (\frac{83}{450})}{2 \ln (0.95)}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x \ln (0.95)}{2 \ln (0.95)} = \frac{\ln (\frac{83}{450})}{2 \ln (0.95)}$

Rewrite the expression.

$\frac{x \ln (0.95)}{\ln (0.95)} = \frac{\ln (\frac{83}{450})}{2 \ln (0.95)}$

Cancel the common factor of $\ln (0.95)$ .

Cancel the common factor.

$\frac{x \ln (0.95)}{\ln (0.95)} = \frac{\ln (\frac{83}{450})}{2 \ln (0.95)}$

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{83}{450})}{2 \ln (0.95)}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (\frac{83}{450})}{2 \ln (0.95)}$

Decimal Form:

$x = 16.47785539 \dots$

Solve for x 1410=4500(0.95)^(2x)+580

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