Question

question 9/24
9 using the equation 1.02^t = 2, solve for t. round to the nearest whole number.
t =

Explanation:

Step1: Take natural log of both sides

To solve \(1.02^t = 2\) for \(t\), we start by taking the natural logarithm (ln) of both sides. This gives us \(\ln(1.02^t)=\ln(2)\). By the logarithm power rule \(\ln(a^b)=b\ln(a)\), we can rewrite the left - hand side as \(t\ln(1.02)=\ln(2)\).

Step2: Solve for t

Now, we solve for \(t\) by dividing both sides of the equation \(t\ln(1.02)=\ln(2)\) by \(\ln(1.02)\). So, \(t = \frac{\ln(2)}{\ln(1.02)}\).
We know that \(\ln(2)\approx0.6931\) and \(\ln(1.02)\approx0.0198\). Then \(t=\frac{0.6931}{0.0198}\approx35\) (when rounded to the nearest whole number).

Answer:

\(35\)