Question

log 125 to the base 5√5

Answer 1

2 is the required answer

Answer 2

Answer:

The value of $log_{5\sqrt{5}} 125$ = 2

Step-by-step explanation:

To find,

$log_{5\sqrt{5}} 125$

Recall the formula

$log_{a^m} a^n = \frac{n}{m}$ ----------------(1)

$a^m Xa^n = a^{m+n}$ -------------(2)

Solution

First let us find the prime factorization of 125

125 = 5 ×5×5 = 5³

125 = = 5³

5√5 = 5 × $5^{\frac{1}{2} }$

by applying equation (2), we get

5 × $5^{\frac{1}{2} }$ = $5^{1+\frac{1}{2} }$ = $5^{\frac{3}{2} }$

5√5 =  = $5^{\frac{3}{2} }$

$log_{5\sqrt{5}} 125$ = $log_{5^\frac{3}{2} }} 5^3$

From the formula(1)

Here a = 5, m = $\frac{3}{2}$ and n = 3

$log_{5\sqrt{5}} 125$ =  $log_{5^\frac{3}{2} }} 5^3$

by applying equation (1) we get

= $\frac{3}{\frac{3}{2} }$  = 3×$\frac{2}{3}$ = 2

Hence the value of $log_{5\sqrt{5}} 125$ = 2

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