Question about finding the exact value using powers of natural number e?

It's asking for the answer in the form of e^P. You just have to find the expression for P

First look at 32:

32 = e^ln(32) -- basic defn of ln

= e^ln(2^5) -- alternate form of 32

= e^(5ln(2)) -- property of ln

now raise this to the 1/ln(2) power

(e^5(ln(2))^(1/ln(2)) = e^(5 ln(2) / ln(2)) -- property of exponents that (a^b)^c = a^(bc)

= e^5 -- simplifying the expression for the exponent.

a million. 5/4 answer: 3^(log base 3 of five/4). word: a^b = e^(b*lna). it extremely is genuine for all complicated numbers a & b. Then 3^(log base 3 of five/4) = e^{ (log base 3 of five/4)*(ln3) } Now replace the backside in log base 3: = e^{ (ln5/4)/ln3 * (ln3) } = e^(ln5/4) = 5/4. an comparable technique could be used for the 2nd decision. d:

ln[32^(1/ln(2))] = (1/ln(2))*ln(32)

ln(32) = ln(2^5) = 5*ln(2)

ln[32^(1/ln(2))] = (1/ln(2))*5*ln(2) = 5

32^(1/ln(2)) = e^5 <<<