Question

(4Q) If c < d and x > 1, which of the following must be true?

Answer 1

Use the rule to change the base

$\log_a(b) = \dfrac{\log_c(b)}{\log_c(a)}$

To express both logarithms in terms of the natural logarithm (for example, any common base would be fine:

$\log_c(x) = \dfrac{\ln(x)}{\ln(c)},\quad \log_d(x) = \dfrac{\ln(x)}{\ln(d)}$

Since the natural logarithm is an increasing function, we have

$c<d\implies \ln(c)>\ln(d)$

which implies

$\dfrac{\ln(x)}{\ln(c)}=\log_c(x)<\log_d(x)=\dfrac{\ln(x)}{\ln(d)}$