a=c^z b=a^x c=b^y
prove xyz=1
Answers
Step-by-step explanation:
Given :-
a=c^z,
b=a^x ,
c=b^y
To find :-
Prove that xyz = 1
Solution :-
Method-1:-
Given that
a=c^z -------(1)
b=a^x -------(2)
c=b^y -------(3)
=> c = (a^x)^y ( from (2))
=> c = a^(xy)
Since (a^m)^n = a^(mn)
=> c = (c^z)^(xy) (from (1))
=> c = c^(xyz)
Since (a^m)^n = a^(mn)
=> c^1 = c^(xyz)
We know that
If the bases are equal then exponents must be equal.
=> 1 = xyz
=> xyz = 1
Hence, Proved.
Method-2:-
Given that
a=c^z
On taking logarithms both sides then
=> log a = log c^z
=> log a = z log c
Since log a^m = m log a
=> z = log a / log c ---------(1)
b=a^x
On taking logarithms both sides then
=> log b = log a^x
=> log b = x log a
Since log a^m = m log a
=> x = log b / log a --------(2)
c=b^y
On taking logarithms both sides then
=> log c = log b^y
=> log c = y log b
Since log a^m = m log a
=> y = log b / log c --------(3)
On multiplying above equations. then
xyz=(log b/log a)×(log b/log c)×(log a/log c)
=> xyz = (log a log b log c)/(log alogblog c)
=> xyz = 1
Hence, Proved.
Used formulae:-
→ (a^m)^n = a^(mn)
→ log a^m = m log a