Answer:
The value of [tex]log_{625}(5)[/tex] is [tex]\frac{1}{4}[/tex]
Step-by-step explanation:
We want to evaluate [tex]log_{625}(5)[/tex].
The base of this logarithm is [tex]625[/tex] and the number is [tex]5[/tex].
We need to express the number [tex]5[/tex] as the base [tex]625[/tex] raised to a certain index.
This implies that, [tex]log_{625}(5)=log_{625}(625^{\frac{1}{4}})[/tex]
Recall now that,
[tex]log_a(m^n)=nlog_a(m)[/tex].
We apply this property to obtain,
[tex]log_{625}(5)=\frac{1}{4}log_{625}(625)[/tex]
Recall again that,
[tex]log_a(a)=1,a\ne0\:or\:1[/tex].This implies that,
[tex]log_{625}(5)=\frac{1}{4}(1)[/tex]
[tex]log_{625}(5)=\frac{1}{4}[/tex]
The correct answer is C.
