Question

Determine the logarithmic function of ln(1+5x)^-5.

(a) 5x + \(\frac{25x^2}{2} + \frac{125x^3}{3} + \frac{625x^4}{4}\) …

(b) x – \(\frac{25x^2}{2} + \frac{625x^3}{3} – \frac{3125x^4}{4}\) …

(c) \(\frac{125x^2}{3} – 625x^3 + \frac{3125x^4}{5}\) …

(d) -25x + \(\frac{125x^2}{2} – \frac{625x^3}{3} + \frac{3125x^4}{4}\) …

This question was addressed to me in quiz.

Query is from Discrete Probability in section Discrete Probability of Discrete Mathematics

Answer 1

Right option is (d) -25x + \(\frac{125x^2}{2} – \frac{625x^3}{3} + \frac{3125x^4}{4}\) …

To explain: Apply the logarithmic law, that is logax = xlog(a). Now the function is ln(1+5x)^-5 = -5log(1+5x). By taking the series = -5(5x – \(\frac{(5x)^2}{2} + \frac{(5x)^3}{3} – \frac{(5x)^4}{4}\) + …) = -25x + \(\frac{125x^2}{2} – \frac{625x^3}{3} + \frac{3125x^4}{4}\) …