What is the value of log(25)(125)?

justaguide | College Teacher | (Level 2) Distinguished Educator

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The value of `log_25 125` has to be determined.

Use the property log a^b = b*log a

To proceed, 125 has to be expressed in terms of 25, it can be seen that 125 = 5^3 and 25 = 5^2, 125 = 25^(3/2)

`log_25 125`

=> `log_25 25^(3/2)`

=> `(3/2)*log_25 25`

Using the property log(a) a = 1 gives `log_25 25 = 1`

=> (3/2)

The required value of `log_25 125 = 1.5`

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to find what is the multiplier of 25, hence you need to divide 125 by 25 such that: `125/25 = 5` .

Hence, you should write `log_25(125) = log_25(5*25)` .

Using the product property of logarithm yields:

`log_25(5*25) = log_25(5) + log_25(25)`

You need to remember that `log_25(25) = 1` .

You need to change the base of `log_25(5)`  such that:

`log_25(5) = 1/(log_5(25))`

You need to find what is the multiplier of 5, hence you need to divide 25 by 5 such that: `25/5 = 5` .

`1/(log_5(25)) = 1/(log_5(5*5)) = 1/(log_5(5)+log_5(5))` ``

`1/(log_5(25)) =1/(1+1) =gt 1/(log_5(25)) = 1/2`

`log_25 (5*25) = 1/2 + 1 =gt log_25 (5*25) = 3/2`

Hence, using properties of logarithms, yields: `log_25 (5*25) = 3/2` .