Question

What is x if `log(7x-12) - 2 log(x)= 1`?

Answer 1

Imaginary Roots

Explanation:

I think roots are imaginary
You may know that `log a^n = n log a`
So, `2 log x = log x^2`
Thus the equation becomes
`log (7x -12) - logx^2 = 1`
Also you may know
`log a - log c = log (a /c)`
Hence the equation reduces to
log `(7x - 12) / x^2 = 1`
You may also know,
if log a to base b is = c, then
`a = b^c`
For `log x` the base is 10
So the equation reduces to
`(7x - 12) / x^2 = 10^1 = 10`
or
`(7x - 12) = 10 * x^2`
ie `10 * x^2 - 7x + 12 = 0`
This is a quadratic equation and the roots are imaginary, since `4 * 10 * 12 > 7^2`