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) = (\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) = (\ln(x))/(\ln(c)),\quad \log_d(x) = (\ln(x))/(\ln(d))`

Since the natural logarithm is an increasing function, we have

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

which implies

`(\ln(x))/(\ln(c))=\log_c(x)<\log_d(x)=(\ln(x))/(\ln(d))`