question\nsolve for $x$ to the nearest 10th.\n$2540 = 310(1.09)^{0.59x} + 640$
question\nsolve for $x$ to the nearest 10th.\n$2540 = 310(1.09)^{0.59x} + 640$
Answer
Explanation:
Step1: Isolate the exponential term
Subtract 640 from both sides. $2540 - 640 = 310(1.09)^{0.59x}$ $1900 = 310(1.09)^{0.59x}$
Step2: Divide by coefficient of exponential
Divide both sides by 310. $\frac{1900}{310} = (1.09)^{0.59x}$ $\approx 6.1290 = (1.09)^{0.59x}$
Step3: Apply natural log to both sides
Take $\ln$ of both sides to bring down exponent. $\ln(6.1290) = \ln\left((1.09)^{0.59x}\right)$ $\ln(6.1290) = 0.59x \cdot \ln(1.09)$
Step4: Solve for x
Rearrange to solve for x, calculate values. $x = \frac{\ln(6.1290)}{0.59 \cdot \ln(1.09)}$ $\ln(6.1290) \approx 1.814$, $\ln(1.09) \approx 0.08618$ $x = \frac{1.814}{0.59 \times 0.08618} = \frac{1.814}{0.05085} \approx 35.7$
Answer:
$35.7$