Please explain this simple rule of logarithms to me Right I know this one is simple and I know that

Keenan Santos 2022-07-06 Answered
Please explain this simple rule of logarithms to me
Right I know this one is simple and I know that I just need a push to make it sink in in my head..
I am studying control systems and in one of the tutorial examples the tutor says
Show that
20 log ( 1 / x ) = 20 log ( x )
I know that when you have a divide or a multiply with logarithms you add them and subtract them but for my own understanding I just need someone to like slowly show me how this works..
If I take the log of the numerator I have 20 log ( 1 ) = 0 but I don't know where to go from here.. So do I now just take the log of the denominator and as the numerator was zero it is just minus whatever the log of the denominator is... Getting myself a bit muddled.. Thanks
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Answers (2)

Caiden Barrett
Answered 2022-07-07 Author has 20 answers
First of all, forget the 20 in front, all you have to do is see that
log ( 1 / x ) = log ( x ) .
You accept the fact that
log ( a b ) = b log ( a )
and use the fact that 1 / x = x 1 , giving you
log ( 1 / x ) = log ( x 1 ) = ( 1 ) log ( x ) = log x
You use the fact that
log ( a / b ) = log a log b
and that log 1 = 0, giving you
log ( 1 / x ) = l o g ( 1 ) log ( x ) = 0 log ( x ) = log ( x )
You use the fact that log 1 = 0, that 1 = x ( 1 / x ) and that
log ( a b ) = log a + log b ,
giving you
0 = log 1 = log ( x ( 1 / x ) ) = log x + l o g ( 1 / x ) .
From the equation
0 = log x + log ( 1 / x ) ,
you get
log ( 1 / x ) = log x
Note that all these derivations can be transformed into each other and are equivalent. I am presenting them all because everybody looks to logarighms in his own way, so make your pick of the favorite.

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Grimanijd
Answered 2022-07-08 Author has 4 answers
Essential for logarithms is equality
g log g a = a
This under the conditions a > 0, g > 0 and g 1
Asking: 'to what power must 3 ( = g) be raised to get 81 ( = a) as outcome?' is the same thing as asking: 'what is the logarithm of 81 on base of 3?' The answer is clearly 4. We have 3 4 = 81 and equivalent is the expression: log 3 81 = 4
Note that:
10 20 log ( 1 x ) = ( 10 log ( 1 x ) ) 20 = ( 1 x ) 20
so the fact that ( 1 x ) 20 = x 20 tells us that 10 is raised in these cases to the same power.
Our conclusion is:
20 log ( 1 x ) = 20 log x
Every rule concerning logarithms can derived likewise. For instance:
g log g a + log g b = g log g a × g log g a = a b = g log g a b
resulting in:
log g a + log g b = log g a b

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Then I take the natural logarithms on both sides;
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subtract ln ( 2 ) on both sides:
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