Question

The coefficient of $x^n$ in the expansion of $\log_a (1 + x)$ is

• A. $\frac{(-1)^{n - 1}}{n}$
• B. $\frac{(-1)^{n - 1}}{n} \log_a \, e$
• C. $\frac{(-1)^{n - 1}}{n} \log_e \, a$
• D. $\frac{(-1)^{n }}{n} \log_a \, e$

Answer 1

Answer: (B)

$\frac{(-1)^{n - 1}}{n} \log_a \, e$

Answer 2

$\log_{a} \left(1+x\right) = \log_{e} \left(1+x\right)\log_{a}e$
$= \log_{a}e \left[\displaystyle\sum_{n=1}^{\infty}\left(-1\right)^{n-1} \frac{X^{n}}{n}\right]$
So, the coefficient of $x^n$ in $\log_a ( 1 + x)$ is
$\frac{\left(-1\right)^{n- 1}}{n} \log_{a}e$