Solve for x 1310=5450(0.88)^(0.16x+2)+510

$1310 = 5450 {(0.88)}^{0.16 x + 2} + 510$

Step 1

Rewrite the equation as $5450 \cdot {0.88}^{0.16 x + 2} + 510 = 1310$ .

$5450 \cdot {0.88}^{0.16 x + 2} + 510 = 1310$

Step 2

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

Subtract $510$ from both sides of the equation.

$5450 \cdot {0.88}^{0.16 x + 2} = 1310 - 510$

Subtract $510$ from $1310$ .

$5450 \cdot {0.88}^{0.16 x + 2} = 800$

Step 3

Divide each term in $(5450 \cdot {0.88}^{0.16 x + 2}) = 800$ by $5450$ and simplify.

Divide each term in $(5450 \cdot {0.88}^{0.16 x + 2}) = 800$ by $5450$ .

$\frac{(5450 \cdot {0.88}^{0.16 x + 2})}{5450} = \frac{800}{5450}$

Simplify the left side.

Cancel the common factor of $5450$ .

Cancel the common factor.

$\frac{5450 \cdot {0.88}^{0.16 x + 2}}{5450} = \frac{800}{5450}$

Divide ${0.88}^{0.16 x + 2}$ by $1$ .

${0.88}^{0.16 x + 2} = \frac{800}{5450}$

Simplify the right side.

Cancel the common factor of $800$ and $5450$ .

Factor $50$ out of $800$ .

${0.88}^{0.16 x + 2} = \frac{50 (16)}{5450}$

Cancel the common factors.

Factor $50$ out of $5450$ .

${0.88}^{0.16 x + 2} = \frac{50 \cdot 16}{50 \cdot 109}$

Cancel the common factor.

${0.88}^{0.16 x + 2} = \frac{50 \cdot 16}{50 \cdot 109}$

Rewrite the expression.

${0.88}^{0.16 x + 2} = \frac{16}{109}$

Step 4

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

$\ln ({0.88}^{0.16 x + 2}) = \ln (\frac{16}{109})$

Step 5

Expand $\ln ({0.88}^{0.16 x + 2})$ by moving $0.16 x + 2$ outside the logarithm.

$(0.16 x + 2) \ln (0.88) = \ln (\frac{16}{109})$

Step 6

Simplify the left side.

Simplify $(0.16 x + 2) \ln (0.88)$ .

Apply the distributive property.

$0.16 x \ln (0.88) + 2 \ln (0.88) = \ln (\frac{16}{109})$

Multiply $\ln (0.88)$ by $0.16$ .

$- 0.02045333 x + 2 \ln (0.88) = \ln (\frac{16}{109})$

Step 7

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

$2 \ln (0.88) - \ln (\frac{16}{109}) = 0.02045333 x$

Step 8

Rewrite the equation as $0.02045333 x = 2 \ln (0.88) - \ln (\frac{16}{109})$ .

$0.02045333 x = 2 \ln (0.88) - \ln (\frac{16}{109})$

Step 9

Divide each term in $0.02045333 x = 2 \ln (0.88) - \ln (\frac{16}{109})$ by $0.02045333$ and simplify.

Divide each term in $0.02045333 x = 2 \ln (0.88) - \ln (\frac{16}{109})$ by $0.02045333$ .

$\frac{0.02045333 x}{0.02045333} = \frac{2 \ln (0.88)}{0.02045333} + \frac{- \ln (\frac{16}{109})}{0.02045333}$

Simplify the left side.

Cancel the common factor of $0.02045333$ .

Cancel the common factor.

$\frac{0.02045333 x}{0.02045333} = \frac{2 \ln (0.88)}{0.02045333} + \frac{- \ln (\frac{16}{109})}{0.02045333}$

Divide $x$ by $1$ .

$x = \frac{2 \ln (0.88)}{0.02045333} + \frac{- \ln (\frac{16}{109})}{0.02045333}$

Simplify the right side.

Simplify each term.

Factor $2$ out of $2 \ln (0.88)$ .

$x = \frac{2 \ln (0.88)}{0.02045333} + \frac{- \ln (\frac{16}{109})}{0.02045333}$

Factor $0.02045333$ out of $0.02045333$ .

$x = \frac{2 \ln (0.88)}{0.02045333 (1)} + \frac{- \ln (\frac{16}{109})}{0.02045333}$

Separate fractions.

$x = \frac{2}{0.02045333} \cdot \frac{\ln (0.88)}{1} + \frac{- \ln (\frac{16}{109})}{0.02045333}$

Divide $2$ by $0.02045333$ .

$x = 97.78354315 \frac{\ln (0.88)}{1} + \frac{- \ln (\frac{16}{109})}{0.02045333}$

Divide $\ln (0.88)$ by $1$ .

$x = 97.78354315 \ln (0.88) + \frac{- \ln (\frac{16}{109})}{0.02045333}$

Move the negative in front of the fraction.

$x = 97.78354315 \ln (0.88) - \frac{\ln (\frac{16}{109})}{0.02045333}$

Multiply by $1$ .

$x = 97.78354315 \ln (0.88) - \frac{1 \ln (\frac{16}{109})}{0.02045333}$

Factor $0.02045333$ out of $0.02045333$ .

$x = 97.78354315 \ln (0.88) - \frac{1 \ln (\frac{16}{109})}{0.02045333 (1)}$

Separate fractions.

$x = 97.78354315 \ln (0.88) - (\frac{1}{0.02045333} \cdot \frac{\ln (\frac{16}{109})}{1})$

Divide $1$ by $0.02045333$ .

$x = 97.78354315 \ln (0.88) - (48.89177157 \frac{\ln (\frac{16}{109})}{1})$

Divide $\ln (\frac{16}{109})$ by $1$ .

$x = 97.78354315 \ln (0.88) - (48.89177157 \ln (\frac{16}{109}))$

Multiply $48.89177157$ by $- 1$ .

$x = 97.78354315 \ln (0.88) - 48.89177157 \ln (\frac{16}{109})$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = 97.78354315 \ln (0.88) - 48.89177157 \ln (\frac{16}{109})$

Decimal Form:

$x = 81.31153456 \dots$

Solve for x 1310=5450(0.88)^(0.16x+2)+510

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