Let , . Consider the function defined for…

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

Let

a>0

a\neq1

. Consider the function

f(x)=a^{\log_{a^3}x}

defined for

x>0

. (i) Express

\log_{a^3}x

in terms of

\log_{a}x

. (ii) Hence show that

f(x)=x^{\tfrac13}

for every

x>0

. (iii) Generalise the result obtained in part (ii) by proving that, for any positive integer

k

a^{\log_{a^{k}}x}=x^{1/k}

for all

x>0

. State clearly any laws of exponents or logarithms that you use.