Logarithm question-I donot know but this question may be solved by any other way also.

Question

Let $(x_0,y_0)$ be the solution of the following equations. $$(2x)^{\ln{2}}=(3y)^{\ln{3}}$$ $$3^{\ln{x}}=2^{\ln{y}}$$

Then $x_0$ is

A) $\frac{1}{6}$

B) $\frac{1}{3}$

C) $\frac{1}{2}$

D) $6$

I have tried this problem by taking log on both sides of the two equations. But, finally I could not make up to get the values of $x$ and $y$.

Answer 1

I would suggest to:

• take ${\ln}$ on both sides of the first equation.
• express ${\ln{y}}$ from the second equation and substitute it into the first equation

Now you have the equation with one variable. After rather simple transformations you will get the answer for ${\ln{x}}$ and later for $x$

Let me know if I'm not clear or you need further help.

Answer 2

Note $\ln 2=a, \ln 3=b,\ln x=u, \ln y=v.$

Logarithm equations are obtained:$$a^2+au=b^2+bv$$ and $$bu=av.$$ With $$v=\frac{b}{a}u$$ find$$u=-a.$$

Conclusion:$$x_0=\frac{1}{2}.$$