Question

X^(logx) = 1000000x Find x.

Answer 1

Answer:

$x = 1000$ or $x = 0.01$

Step-by-step explanation:

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

Given,

$x^{\log x} = 1000000x$

⇒ $x^{\log x} = 10^{6} \times x$

Now, taking log on both sides, we get

$\log x^{\log x} = \log 10^{6} \times x$

⇒ $\log x .\log x = \log 10^{6} + \log x$

{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 $x = 10^{- 2} = \frac{1}{100} = 0.01$ (Answer)