Use the rule to change the base
[tex]\log_a(b) = \dfrac{\log_c(b)}{\log_c(a)}[/tex]
To express both logarithms in terms of the natural logarithm (for example, any common base would be fine:
[tex]\log_c(x) = \dfrac{\ln(x)}{\ln(c)},\quad \log_d(x) = \dfrac{\ln(x)}{\ln(d)}[/tex]
Since the natural logarithm is an increasing function, we have
[tex]c<d\implies \ln(c)>\ln(d)[/tex]
which implies
[tex]\dfrac{\ln(x)}{\ln(c)}=\log_c(x)<\log_d(x)=\dfrac{\ln(x)}{\ln(d)}[/tex]
