To determine when the motorcycle will be worth $7000, we will set the depreciation function equal to $7000:
\[ f(x) = 14000(1 - 0.09)^x = 7000 \]
This simplifies to:
\[ 14000(0.91)^x = 7000 \]
Next, we can divide both sides by 14000:
\[ (0.91)^x = \frac{7000}{14000} = 0.5 \]
Now, we will take the logarithm of both sides to solve for \(x\):
\[ \log((0.91)^x) = \log(0.5) \]
Using the power rule of logarithms:
\[ x \log(0.91) = \log(0.5) \]
Now, solving for \(x\):
\[ x = \frac{\log(0.5)}{\log(0.91)} \]
Using a calculator to find the values of the logarithms:
\[ \log(0.5) \approx -0.3010 \] \[ \log(0.91) \approx -0.0434 \]
Now we can substitute these values into the equation:
\[ x = \frac{-0.3010}{-0.0434} \approx 6.93 \]
Rounding to one decimal place, we find that \(x \approx 6.9\) years.
Given the options, the motorcycle will be worth $7000 in approximately 7 years.
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