Question

Solve for x the equation 2^x + 3 = 5^x + 2.

(a) ln (24/8)

(b) ln (25/8) / ln (2/5)

(c) ln (32/5) / ln (2/3)

(d) ln (3/25)

I had been asked this question in my homework.

The doubt is from Discrete Probability in division Discrete Probability of Discrete Mathematics

Answer 1

Right choice is (b) ln (25/8) / ln (2/5)

Easiest explanation: Given that 2^x + 3 = 5^x + 2. By taking ln of both sides: ln (2^x + 3) = ln (5^x + 2)

⇒  (x + 3) ln 2 = (x + 2) ln 5

⇒  x ln 2 + 3 ln 2 = x ln 5 + 2 ln 5

⇒  x ln 2 – x ln 5 = 2 ln 5 – 3 ln 2

⇒ x = ( 2 ln 5 + 3 ln 2 ) / (ln 2 – ln 5) = ln (5^2 / 2^3) / ln (2/5) = ln (25/8) / ln (2/5).