To find out when the motorcycle will be worth $7000, we need to solve the equation:
\[ f(x) = 14000(1 - 0.09)^x = 7000 \]
First, let's set the equation:
\[ 14000(1 - 0.09)^x = 7000 \]
Now, divide both sides by 14000:
\[ (1 - 0.09)^x = \frac{7000}{14000} \]
This simplifies to:
\[ (1 - 0.09)^x = 0.5 \]
Now, calculate \(1 - 0.09\):
\[ 1 - 0.09 = 0.91 \]
So, we have:
\[ 0.91^x = 0.5 \]
To solve for \(x\), we can take the logarithm of both sides:
\[ \log(0.91^x) = \log(0.5) \]
Using the power rule of logarithms:
\[ x \cdot \log(0.91) = \log(0.5) \]
Now, isolate \(x\):
\[ x = \frac{\log(0.5)}{\log(0.91)} \]
Now, calculate the values:
Using a calculator, we find:
\[ \log(0.5) \approx -0.3010 \] \[ \log(0.91) \approx -0.0436 \]
Now, substitute these values into the equation:
\[ x \approx \frac{-0.3010}{-0.0436} \approx 6.90 \]
Therefore, \(x\) is approximately 6.90. Since \(x\) represents the number of years, it means that the motorcycle will be worth $7000 after about 7 years.