ln and e^x are inverse functions, so ln(e^x) = x for any x. Therefore ln(e²) = 2. A common error is thinking the answer involves e itself, confusing the base with the exponent the logarithm returns.
Question 2 True / False
The expression ln(e) equals e, since the natural log and e are closely related.
TTrue
FFalse
Answer: False
ln(e) = 1, not e. The logarithm asks: 'e to what power gives me this input?' Since e¹ = e, the answer is 1. A logarithm always returns an exponent — a number, never the base itself.
Question 3 Short Answer
A savings account compounds interest continuously at 5% per year. Write the formula for the balance A after t years if the initial principal is A₀.
Think about your answer, then reveal below.
Model answer: A = A₀ · e^(0.05t)
Continuous compounding uses A = A₀ · e^(kt), where k is the annual rate as a decimal. Here k = 0.05. This formula arises because e is defined as the limit of (1 + 1/n)^n as n → ∞, capturing what happens when compounding becomes infinitely frequent.