Question

I just need to see whether I'm correct or not.

If
`log_(3)\ x^(2)=4` and
`log_(2)\ y^(3)=6`, and
`log_(b)\ x+log_(b)\ y=(1)/(2)`, where x, y > 0, then the value of b is ____

I got b = 1296 but the answer key gave b = 6.

Who's correct?

Answer 1

If
`log_3(x^2)=4`, then that means
`3^4=x^2`, so
`x=9`.

If
`log_2(y^3)=6`, then that means
`2^6 = y^3`, so
`y=4`.

`log_b(x)+log_b(y)=log_b(x\cdot y) = log_b(36)`

so, the last statement is equivalent to
`log_b(36) = (1)/(2)` or
`b^(1/2)=36`.

This means that
`b=36^2 = 1296`.

I'm on board with your answer.

If it had been
`log_b(x)+log_b(y)=2`, then
`b=6` would have been correct.