question\nsolve for $x$ to the nearest 10th.\n$2100 = 920(1.06)^{\frac{x}{8}} + 600$

Answer:

$x \approx 10.6$

Explanation:

Step1: Isolate the exponential term

Subtract 600 from both sides: $2100 - 600 = 920(1.06)^{\frac{x}{8}}$ $1500 = 920(1.06)^{\frac{x}{8}}$

Step2: Divide to simplify

Divide both sides by 920: $\frac{1500}{920} = (1.06)^{\frac{x}{8}}$ $1.6304 \approx (1.06)^{\frac{x}{8}}$

Step3: Apply natural logarithm

Take $\ln$ of both sides: $\ln(1.6304) = \ln\left((1.06)^{\frac{x}{8}}\right)$ Use logarithm power rule $\ln(a^b)=b\ln(a)$: $\ln(1.6304) = \frac{x}{8}\ln(1.06)$

Step4: Solve for x

Rearrange and calculate: $x = 8 \times \frac{\ln(1.6304)}{\ln(1.06)}$ $x = 8 \times \frac{0.488}{0.0583}$ $x \approx 8 \times 1.327$ $x \approx 10.6$