Question

If 2^ = 3^ = 6^ , then find the relation between a, b and c.

Answer 1

Correct question:

If 2^a = 3^b = 6^c , then find the relation between a , b and c .

Answer:

1/c = 1/a + 1/b

Note:

★ a^m × a^n = a^(m+n)

★ a^m / a^n = a^(m - n)

★ a^m × b^m = (a×b)^m

★ a^m / b^m = (a/b)^m

★ a^m = a^n => m = n

★ a^m = b => a = b^(1/m)

Solution:

Let 2^a = 3^b = 6^c = k

Thus,

If 2^a = k , then

2 = k^(1/a)

If 3^b = k , then

3 = k^(1/b)

If 6^c = k , then

6 = k^(1/c)

Now,

=> 6 = k^(1/c)

=> 2×3 = k^(1/c)

=> [ k^(1/a) ] × [ k^(1/b) ] = k^(1/c)

=> k^(1/a + 1/b) = k^(1/c)

=> 1/a + 1/b = 1/c

=> 1/c = 1/a + 1/b

Hence,

Required answer is ;

1/c = 1/a + 1/b