Solve for x 3420=3270(0.97)^(2x-5)+920

$3420 = 3270 {(0.97)}^{2 x - 5} + 920$

Rewrite the equation as $3270 \cdot {0.97}^{2 x - 5} + 920 = 3420$ .

$3270 \cdot {0.97}^{2 x - 5} + 920 = 3420$

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

Subtract $920$ from both sides of the equation.

$3270 \cdot {0.97}^{2 x - 5} = 3420 - 920$

Subtract $920$ from $3420$ .

$3270 \cdot {0.97}^{2 x - 5} = 2500$

Divide each term in $(3270 \cdot {0.97}^{2 x - 5}) = 2500$ by $3270$ and simplify.

Divide each term in $(3270 \cdot {0.97}^{2 x - 5}) = 2500$ by $3270$ .

$\frac{(3270 \cdot {0.97}^{2 x - 5})}{3270} = \frac{2500}{3270}$

Simplify the left side.

Cancel the common factor of $3270$ .

Cancel the common factor.

$\frac{3270 \cdot {0.97}^{2 x - 5}}{3270} = \frac{2500}{3270}$

Divide ${0.97}^{2 x - 5}$ by $1$ .

${0.97}^{2 x - 5} = \frac{2500}{3270}$

Simplify the right side.

Cancel the common factor of $2500$ and $3270$ .

Factor $10$ out of $2500$ .

${0.97}^{2 x - 5} = \frac{10 (250)}{3270}$

Cancel the common factors.

Factor $10$ out of $3270$ .

${0.97}^{2 x - 5} = \frac{10 \cdot 250}{10 \cdot 327}$

Cancel the common factor.

${0.97}^{2 x - 5} = \frac{10 \cdot 250}{10 \cdot 327}$

Rewrite the expression.

${0.97}^{2 x - 5} = \frac{250}{327}$

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

$\ln ({0.97}^{2 x - 5}) = \ln (\frac{250}{327})$

Expand $\ln ({0.97}^{2 x - 5})$ by moving $2 x - 5$ outside the logarithm.

$(2 x - 5) \ln (0.97) = \ln (\frac{250}{327})$

Simplify the left side.

Apply the distributive property.

$2 x \ln (0.97) - 5 \ln (0.97) = \ln (\frac{250}{327})$

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

$2 x \ln (0.97) - 5 \ln (0.97) - \ln (\frac{250}{327}) = 0$

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

Add $5 \ln (0.97)$ to both sides of the equation.

$2 x \ln (0.97) - \ln (\frac{250}{327}) = 5 \ln (0.97)$

Add $\ln (\frac{250}{327})$ to both sides of the equation.

$2 x \ln (0.97) = 5 \ln (0.97) + \ln (\frac{250}{327})$

Divide each term in $2 x \ln (0.97) = 5 \ln (0.97) + \ln (\frac{250}{327})$ by $2 \ln (0.97)$ and simplify.

Divide each term in $2 x \ln (0.97) = 5 \ln (0.97) + \ln (\frac{250}{327})$ by $2 \ln (0.97)$ .

$\frac{2 x \ln (0.97)}{2 \ln (0.97)} = \frac{5 \ln (0.97)}{2 \ln (0.97)} + \frac{\ln (\frac{250}{327})}{2 \ln (0.97)}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x \ln (0.97)}{2 \ln (0.97)} = \frac{5 \ln (0.97)}{2 \ln (0.97)} + \frac{\ln (\frac{250}{327})}{2 \ln (0.97)}$

Rewrite the expression.

$\frac{x \ln (0.97)}{\ln (0.97)} = \frac{5 \ln (0.97)}{2 \ln (0.97)} + \frac{\ln (\frac{250}{327})}{2 \ln (0.97)}$

Cancel the common factor of $\ln (0.97)$ .

Cancel the common factor.

$\frac{x \ln (0.97)}{\ln (0.97)} = \frac{5 \ln (0.97)}{2 \ln (0.97)} + \frac{\ln (\frac{250}{327})}{2 \ln (0.97)}$

Divide $x$ by $1$ .

$x = \frac{5 \ln (0.97)}{2 \ln (0.97)} + \frac{\ln (\frac{250}{327})}{2 \ln (0.97)}$

Simplify the right side.

Cancel the common factor of $\ln (0.97)$ .

Cancel the common factor.

$x = \frac{5 \ln (0.97)}{2 \ln (0.97)} + \frac{\ln (\frac{250}{327})}{2 \ln (0.97)}$

Rewrite the expression.

$x = \frac{5}{2} + \frac{\ln (\frac{250}{327})}{2 \ln (0.97)}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{2} + \frac{\ln (\frac{250}{327})}{2 \ln (0.97)}$

Decimal Form:

$x = 6.90752198 \dots$

Solve for x 3420=3270(0.97)^(2x-5)+920

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