Question

X^(logx) = 1000000x
Find x.

Answer 1

`x = 1000` or
`x = 0.01`

Explanation:

We are given a logarithmic equation of x and we have to solve it for x.

Given,

`x^(\log x) = 1000000x`

⇒
`x^(\log x) = 10^(6) * x`

Now, taking log on both sides, we get

`\log x^(\log x) = \log 10^(6) * x`

⇒
`\log x .\log x = \log 10^(6)  + \log x`

{Since
`\log a^(b) = b\log a` and
`\log ab = \log a + \log b`}

⇒
`(\log x)^(2) = 6 + \log x`

{Since log 10 = 1}

⇒ a² = 6 + a {Where, a = log x}

⇒ a² - a - 6 = 0

⇒ (a - 3)(a + 2) = 0

⇒ a = 3 or a = -2

⇒
`\log x = 3` or
`\log x = - 2`

Now, converting logarithm to exponential form, we get,

⇒
`x = 10^(3) = 1000` or
`x = 10^(- 2) = (1)/(100) = 0.01` (Answer)