Answer:
x=25
Step-by-step explanation:
The actual question is shown in the image attached.
Solve for x.
sqrt(x)^(log_5(x) -1) = 5 ......................(1)
Solution:
from (1)
sqrt(x)^(log_5(x) -1) = 5 .......separate exponents, law of log a^(b-c)=a^b/a^c
sqrt(x)^log_5(x) / sqrt(x) = 5 .......... cross multiply
sqrt(x)^log_5(x) = 5sqrt(x) .............. square both sides
(sqrt(x)^log_5(x))^2 = 25x .............. modify base a^(2b) = (a^2)^b
(sqrt(x)^2)^log_5(x) = 25x
x^log_5(x) = 25x .................... take log_5 on both sides
log_5(x) * log_5(x) = log_5(5^2*x) ............... simplify RHS
log_5(x) * log_5(x) = log_5(25)+log_5(x)
log_5(x) * log_5(x) = 2+log_5(x) ........ simplify
log_5(x) ^2 -log_5(x) -2 = 0 ........... substitute y = log_5(x)
y^2 - y -2 = 0
(y-2)(y+1) = 0
y=2 or
y = -1 ................... y = log_5(x) >= 0 , y=-1 rejected
y = 2
log_5(x) = 2
raise to base of 5
5^log_5(x) = 5^2
x = 25
Check by substituting x = 25 in (1)
sqrt(x)^(log_5(x) -1)
= sqrt(25)^(log_5(25) -1)
= 5^(2-1)
= 5 equal RHS, therefore solution is correct.
