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Logarithms and Substitution

Source: AHSME 1963 Problem 30

January 11, 2014
logarithmsfunctionAMC

Problem Statement

Let F=log⁡1+x1−x.F=\log\dfrac{1+x}{1-x}. Find a new function GG by replacing each xx in FF by 3x+x31+3x2,\dfrac{3x+x^3}{1+3x^2}, and simplify. The simplified expression GG is equal to:
<spanclass=′latex−bold′>(A)</span> −F<spanclass=′latex−bold′>(B)</span> F<spanclass=′latex−bold′>(C)</span> 3F<spanclass=′latex−bold′>(D)</span> F3<spanclass=′latex−bold′>(E)</span> F3−F<span class='latex-bold'>(A)</span>\ -F \qquad <span class='latex-bold'>(B)</span>\ F\qquad <span class='latex-bold'>(C)</span>\ 3F \qquad <span class='latex-bold'>(D)</span>\ F^3 \qquad <span class='latex-bold'>(E)</span>\ F^3-F
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