It is important to note that both 125 and 25 are powers of 5. 5 to the third yields 125, 5 squared yields 25.
By definition of the logarithm, the number we are looking for is the number x for which [tex]125^x=25[/tex] holds. Using the above facts, we get:
[tex](5^3)^x=5^2 \\ 5^{3x}=5^2[/tex].
Since the bases of the exponents are equal, we can equate the exponents. 3x=2. Hence, x=2/3. This the answer. [tex]log_{125} (25)= \frac{2}{3} [/tex]
