Solving an Equation That Can Be Simplified to the Form y = Ae^kt

Question

The math problem:

And this is my solution:

And the book's solution is:

I tried using a math website (Mathway) to see what solution they would get and they got the same as mine:

Is the book incorrect?

Answer 1

The book is not incorrect, it's the same answer just expressed differently. The quotient rule for logarithms states that $\log_{b}\left(\frac{x}{y}\right) = \log_{b}(x) - \log_{b}(y)$. If $x = 1$ as it does in $\ln\left(\frac{1}{2}\right)$ then the quotient rule would tell you that $\ln\left(\frac{1}{2}\right) = -\ln(2)$ as $\ln(1) = 0$. Thus, $\frac{1}{2}\ln\left(\frac{1}{2}\right) = -\frac{1}{2}\ln\left(2\right)$.