Defining the Range of Variables in Logarithmic and Radical Expressions

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Homework Statement



Real variables x and y are related by the equation

ln(2+y) = 5ln(3 - x) - 2 sqrt x ...(sorry, I haven't as yet got the hang on LaTeX)

Determine the range of values of x and y for which the expressions on each side of this equation are defined.


Homework Equations





The Attempt at a Solution



I haven't really been able to make an attempt at a solution. I think I have to take the exp of each side, but I am not sure excatly what way I go about this, so if someone could give me some advice, that would be fantastic.

Thanks in advance.

Sean
 
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HINTS:

(1) For what values of \gamma is \ln\gamma defined?

(2) For what values of \eta is \sqrt{\eta} defined?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
View full post »

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