Question

Let

$F=\log\frac{1+x}{1-x}.$

Form a new function $G$ by replacing each $x$ in $F$ by

$\frac{3x+x^3}{1+3x^2},$

and simplify. The simplified expression $G$ is equal to:

$-F$

$F$

$3F$

$F^3$

$F^3-F$

Answer: C

Concepts:logarithm factoring algebraic manipulation

Difficulty rating: 2020

Small Hint:

Call the substituted fraction $u$ and simplify $\frac{1+u}{1-u}$

Big Hint:

Its numerator and denominator factor as cubes

Solution:

For $u=\frac{3x+x^3}{1+3x^2},$ $\begin{aligned} \frac{1+u}{1-u} &=\frac{1+3x+3x^2+x^3} {1-3x+3x^2-x^3}\\ &=\left(\frac{1+x}{1-x}\right)^3. \end{aligned}$ Therefore $G=\log(\frac{1+x}{1-x})^3,$ so $G=3F.$

Thus, the correct answer is C.

1963 AMC 12 Problem 30

Answer: C

Difficulty rating: 2020

Small Hint:

Big Hint:

Solution:

Problem 30 in Other Years