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MATHEMATICS · Multi-part · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

Let

f(x)=\log_b x

and

g(x)=\log_c x

with

b,c>1

and

b\neq c

. (a) Prove that if

f(x)>g(x)

for every

x>1

, then

b<c

. (b) Deduce that for

0<x<1

the inequality is reversed, that is,

f(x)<g(x)

. (c) For the particular pair

f(x)=\log_5 x

and

g(x)=\log_7 x

, determine all real numbers

x

for which

f(x)-g(x)

is positive, negative, or zero.