Logarithm Question....Need some major help!!!?

Question

I am taking a history of math class, and one of my questions has me stumped. The questions is:

Show that: log(base N) b = log(base 1/N) (1/b)

The whole class doesn't really make sense to me at all.

Answer 1

what a dumb contrived problem they gave you.

b = N ^ log[N] b = (1/N)^ log[1/N] b
= N^ -log[1/N] b = N^log[1/N](1/b)
so log[N]b = log[1/N](1/b)

Answer 2

Let log(base N) b = x. Then b = N^x and 1/b = 1/(N^x).

Now, let log(base 1/N) 1/b = y. Then 1/b = (1/N)^y = 1/(N^y).

But, in the first line, we stated that 1/(N^x) = 1/b.

Therefore, 1/(N^x) = 1/b = 1/(N^y) and, by transitivity, 1/(N^x) = 1/(N^y), which forces x to be equal to y.

Also, since x = log(base N) b and y = let log(base 1/N) 1/b, we conclude that log(base N) = log(base 1/N) 1/b, which was to be shown.

Answer 3

The formula for changing base is

log(baseN)b =( log (base 1/N) b)/log(base 1/N) N

but log(base 1/N) N = -1 as (1/N) ^-1 =N
so log(base N )b = - log (base 1/N)b = log (base(1/N) (1/b)as log(1/b) always = log 1 -log b and log 1 =0

Answer 4

Is very simple actually, all you need to understand that the log of a number is an exponent. We could rewrite it as what exponent would make N into b that is equal to the exponent that would make 1/N into 1/b. So its proven that 1 to any power is itself. that means that if N^x = b then (1/N)^x = (1/b)

Answer 5

it is known that if log(base N)b=X
then N^X=b
or (1/N)^X=1/b
or log(base 1/N)(1/b)=X=log(base N)b

Answer 6

log(baseN)b
=
log(base10)b/log(base10)N
=
(-log(base10)b)
/(-log(base10)N)
=
log(base10)(1/b)
/log(base10)(1/N)
=
log(base 1/N) (1/b)