Question

solve the exponential equation by taking the logarithm of both sides. express your answer in exact simplest form.
$e^{0.02k} + 2 = 59$
part: 0 / 2
part 1 of 2
subtract 2 from both sides to isolate the exponential expression.
$e^{0.02k} = 57$
the exponential expression has a base of $e$, so it is convenient to take the natural logarithm of both sides.
$ln e^{0.02k} = ln 57$
use the power property of logarithms.
$square ln square = ln 57$

Explanation:

Step1: Recall Power Property of Logarithms

The power property of logarithms states that $\ln a^b = b\ln a$. For the left - hand side $\ln e^{0.02k}$, using this property, we can rewrite it as $0.02k\ln e$. Since $\ln e = 1$, the left - hand side simplifies to $0.02k$. But in the form required by the problem ( $\square\ln\square=\ln57$), we know that $\ln e^{0.02k}=0.02k\ln e$. So the first box should be $0.02k$ and the second box should be $e$.

Answer:

The first box is $0.02k$ and the second box is $e$, so the equation is $0.02k\ln e=\ln57$ (and since $\ln e = 1$, we can also think of it as $0.02k=\ln57$ for the next step of solving for $k$). In the form $\square\ln\square=\ln57$, the first $\square$ is $0.02k$ and the second $\square$ is $e$.