## vivalakoda 2 years ago I need help with an IMPOSSIBLE equation?

1. Troublemaker

Well what is it?...

2. vivalakoda

3 log 2 x + 1/2 log 2 y – 3 log 2 z = log 2 (x^3√y / z^3).

3. ParthKohli

Nothing is impossible.

4. ParthKohli

Okay, that's an impossible equation indeed. But it has multiple solutions.

5. vivalakoda

Exaggeration of course, ha

6. vivalakoda

So it is capable of being solved? I could call it true?

7. ParthKohli

Yes, it can be solved by using these identities:$\log_a b + \log _a c = \log_a (bc)$and$\log_a b - \log_a c = \log_a (b/c)$

8. jiteshmeghwal9

$\log_2x^3+\log_2y^{1/2}-\log_2z^3=\log_2(x^3\sqrt{y}/z^3)$

9. ParthKohli

Two more:$a\log b = \log b^a$and$\log_a b = \log_a c \iff b = c$

10. jiteshmeghwal9

now i think it's possible now

11. ParthKohli

I am very, very lazy and I am very, very serious about that. So I think @jiteshmeghwal9 will continue helping :-p

12. jiteshmeghwal9

o_O

13. vivalakoda

It asks me if the equation is true, and if so then to explain the properties used. o.o

14. ParthKohli

The identities I listed.

15. ParthKohli

Use them one-by-one.

16. jiteshmeghwal9

$$log_2x^3+log_2(y^{1/2})=log_2(x^3\sqrt{y})$$ now $$log_2(x^3\sqrt{y})-log_2z^3=log_2(x^3\sqrt{y}/z^3)$$

17. ParthKohli

That's it, right there. ^

18. jiteshmeghwal9

$\log_2x^3+\log_2y^{1/2}-\log_2z^3=\log_2(x^3\sqrt{y}/z^3)$since$\log_2x^3+\log_2(y^{1/2})=\log_2(x^3\sqrt{y})$$\log_2(x^3\sqrt{y})-\log_2z^3=\log_2(x^3\sqrt{y}/z^3)$so,$\log_2(x^3\sqrt{y}/z^3)=\log_2(x^3\sqrt{y}/z^3)$H.P.

19. ParthKohli

Brotip: Use Q.E.D. instead of H.P. :-)

20. jiteshmeghwal9

Q.E.D=?

21. vivalakoda

I am so confused. Thank you everyone for the help, I'll just do my best to take the identities you both listed and write something about it. Very much appreciated!

22. jiteshmeghwal9

yw :) Best of luck;)

23. vivalakoda

By properties they mean the logarithmic properties :c I just asked my teacher. So the power property, the product property, and the quotient property? Would any of those fit?

24. AravindG

Power Property $\log a^b=b \log a$ Product Property $\log ab= \log a +\log b$ Quotient property $\log \dfrac{a}{b}=\log a - \log b$

25. jiteshmeghwal9

$$blog_ac=log_ac^b$$ $$log_ab+log_ac=log_abc$$ $$log_ab - log_ac=log_a\dfrac{b}{c}$$

26. jiteshmeghwal9

these are the only properties used in the question

27. vivalakoda

How do they show that the equation is true, though?

28. jiteshmeghwal9

I have proved this above