Question

How can I find x?
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Answer 1

\[\log_{2}x = \log_{1/2}x \]

Answer 2

Change the bases.

Answer 3

\[\frac{ \log_{10}x }{ \log_{10}2{} }=\frac{ \log_{10}x }{\log_{10}0.5 }\]

Answer 4

And now? What should I do?

Answer 5

Since now you have the same base, you can use the addition and subtraction rule for logs.

Answer 6

\[\frac{ logx}{ \log2 }-\frac{ logx }{ \log0.5 }=0\]
How is the substraction rule? this one?
\[\log \frac{ x }{ y }=logx-logy\]

Answer 7

Oh hmm I forgot about that.

Answer 8

i think maybe it is best to do this by inspection
the only way two logs are going to be the same is if \(x=1\) making both sides 0

Answer 9

by the change of base formula, one logarithm is a constant multiple of the other
i.e. \(\log_b(x)=c\log_a(x)\) and the only way for that to happen is if they are both zero

Answer 10

sort of like saying "solve for \(z\) if \(3z=z\) it is pretty clear that \(z=0\) is the only solution, so \(\log_2(x)=0\) making \(x=1\)

Answer 11

But if I did not know it, how could I know x=1?

Answer 12

So are you saying we should equate both sides to 0? What if we change them to index form?

Answer 13

How? A little help please

Answer 14

I'm not sure I'm asking Satellite myself. But, what happens if we use the first definition from http://www.mathwords.com/l/logarithm_rules.htm ?

Answer 15

But this are not single logarithms. Let's try

\[2^{\log_{1/2}x }=x\]

Maybe, we should be sure that both have base two.
\[\log_{2}x=\frac{ \log_{2}x }{ \log_{2}0.5 } \]

So...\[2^{\frac{ \log_{2}x }{ \log_{2}0.5 } }= x\]


I dont know, I am completely lost

Answer 16

what i am saying in english is this: the change of base formula tells you that
\[\log_b(x)=\frac{\log_a(x)}{\log_a(b)}\] which is the same thing as saying
\[\log_a(b)\times \log_b(x)=\log_a(x)\]
don't forget \(\log_a(b)\) is not a function, it is number (constant)
this means every log is a constant multiple of every other log

Answer 17

you have two different logs, but they are the same except for a constant
it is like being asked to solve
\[3z=z\] for \(z\)
what do you get in that case?

Answer 18

Ok, I understandnow! Thanks!

Answer 19

whew
you were doing way way too much work
the only way for \(3z=z\) is if \(z=0\) meaning \(\log_2(x)=0\) forcing \(x=1\)

Answer 20

bty it is always the case that
\[\log_b(x)=-\log_{\frac{1}{b}}(x)\] again by the change of base formula
so in this case it is more like being asked to solve \(z=-z\) which of course still means \(z=0\)

Answer 21

Thanks a lot. Thanks, wolfe8

Answer 22

I didn't help

Answer 23

You tried :)