How to Solve Complex Trigonometry Equations?

[vijay123]

see, here's how i did it...2^log2/3x^log2=(9x)^log3
((2^log2)/(3^log3))^(1/log6)=3x
2^log(6)2/3^log(6)3=3x
2/3=3x
x=2/9

 

[vijay123]

sry mist some steps...but u ll get the picture once you do it yourself...you think the method is correct?...the ans. is correct coz i checked it in my graphing calc.

 

[VietDao29]

Uhm, sorry for misreading the problem. But by the way, you should use parentheses to make the expressiona clearer. Saying:

b) (2/3x)^(log2) = (9x)^(log3)

is a bit confusing. Since, there are 2 ways to read it.
Is it:
$$\left( \frac{2}{3} x \right) ^ {\log 2} = (9x) ^ {\log 3}$$
or:
$$\left( \frac{2}{3x} \right) ^ {\log 2} = (9x) ^ {\log 3}$$
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Your work looks good to me. The method is correct. Just missing a few steps. Congratulations. :)
By the way, do you know how to show that:
$$\frac{2 ^ {\log_6 2}}{3 ^ {\log_6 3}} = \frac{2}{3}$$?

 

[vijay123]

yes...we have learned how to prove it...thanks for your help

 

[vijay123]

omg...how do you show that 2^(log2/log6)=2, by the case above...
i have only learned that 2^(log6/log2)=6!

 

[VietDao29]

omg...how do you show that 2^(log2/log6)=2, by the case above...

No, this is not correct.
$$2 ^ {\frac{\log 2}{\log 6}} = 2 ^ {\log_6 2} \neq 2$$!
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In general, you can show that:
$$\frac{a ^ {\log_{ab}a}}{b ^ {\log_{ab}b}} = \frac{a}{b}$$.
Since the base of the log in both numerator and denominator is ab. So, to apply the property of inverse function: f(f^-1(x)) = x, or f^-1(f(x)) = x, i.e $$(ab) ^ {\log_{ab} x} = x$$.
We should first multiplying both numerator, and denominator by either $$b ^ {\log_{ab}a}$$ or $$a ^ {\log_{ab}b}$$ to make it have the form above.
So:
$$\frac{a ^ {\log_{ab}a}}{b ^ {\log_{ab}b}} = \frac{\left( a ^ {\log_{ab}a} \right) \left( a ^ {\log_{ab}b} \right)}{\left( b ^ {\log_{ab}b} \right) \left( a ^ {\log_{ab}b} \right)} = \frac{a ^ {\log_{ab} a + \log_{ab} b}}{\left( ab \right) ^ {\log_{ab} b}} = ...$$
Apply this to your problem, Note that: 6 = 2 . 3
Can you go from here? :)