Question

if a^x=b b^y=c c^z=a , prove that x=1/yz​

Answer 1

Answer:

Samajh nahi aaya

Step-by-step explanation:

Kuck bhi konsi class ka hai

Answer 2

Step-by-step explanation:

Given :-

a^x=b

b^y=c

c^z=a

To find :-

Prove that x=1/yz

Solution :-

Method-1:-

Given that

a^x=b --------(1)

b^y=c ---------(2)

c^z=a ----------(3)

=> (b^y)^z = a ( from (2))

=> b^(yz) = a

Since (a^m)^n = a^(mn)

=> (a^x)^(yz) = a (from (1))

=> a^(xyz) = a

Since (a^m)^n = a^(mn)

=> a^(xyz) = a^1

=> xyz = 1

Since the bases are equal then exponents must be equal.

=>x = 1/yz

Hence, Proved.

Method -2:-

a^x=b

On taking logarithms both sides then

=> log a^x = log b

=> x log a = log b

Since log a^m = m log a

=> x = log b / log a ----------------(1)

b^y=c

On taking logarithms both sides then

=> log b^y = log c

=> y log b = log c

Since log a^m = m log a

=> y = log c / log b ----------------(2)

and

c^z=a

On taking logarithms both sides then

=> log c^z = log a

=> z log c = log a

Since log a^m = m log a

=> z = log a / log c ----------------(3)

On multiplying (1),(2)&(3)

=> x×y×z

= (log b / log a)×(log c / log b)×(log a / log c)

=> xyz =( log a log b log c )/ (log a log b log c)

=> xyz = 1

=> x = 1/yz

Hence , Proved.

Answer:-

If a^x=b b^y=c c^z=a then x=1/yz

Used formulae:-

• (a^m)^n = a^(mn)

• If the bases are equal then exponents must be equal.

• log a^m = m log a