Is the only way to solve n = (log 0.05) / (log -0.6478) that has negative log numbers is only guessing and can

Question

Is the only way to solve n = (log 0.05) / (log -0.6478) that has negative log numbers is only guessing and can we just remove the negative sign for the number? Is it correct and what is the reason behind it?

The context comes from my previous question which is below:
1 - r^n = 0.95
r^n = 0.05
n log r = log 0.05
n = (log 0.05) / (log -0.6478)

Oops -- log of a negative number!! Will have to think about this some more. But if I say n = (log 0.05) / (log 0.6478) (taking out the minus sign), then I get n = 6.8999, rounded up to n = 7.
IS IT CORRECT?

Out of curiosity, we have

S(7) = a(1-r^7)/(1-r) = 0.6359 a
S = a/(1 - r) = 0.6069

and S(7) / S = 1.04787

so S(7) is within 5% of S. Looks like your answer is n=7.

My only problem involves the log of a negative number, but on that topic, my brain is foggy right now. Anyway, hope this helps.

ANYONE has other methods of doing other than guessing method?

sorry, I couldn't understand the negative sign removal part

Answer 1

That was my answer you quoted above, and when I encountered the "log of a negative number," I did just make a willy-nilly guess, so in my opinion, your characterization was unwarranted. (In fact, you had asked me to look at this problem in the first place.)

When I removed the negative sign, I obtained an answer, and, if you looked at the "sources" section, you'd have seen that it was the right answer.

In light of your denigration of what you called my "guessing method," I'll add the following (which I felt I didn't have to research in light of the correct answer I obtained and checked) ...

The logarithms could just as easily have been natural or Napierian logs instead of common logs. Instead of
n = (log 0.05) / (log -0.6478), I could have written
n = (ln 0.05) / (ln -0.6478).

But ln(-0.6478) = (ln -1) + (ln 0.6478)
which begs the question, what is ln(-1)?

To answer this, we turn to Euler's famous identity,

e^(i pi) + 1 = 0
e^(i pi) = -1
(i pi) ln e = ln(-1)

But since ln e = 1, we have

ln(-1) = i pi

which is entirely in the imaginary portion of the complex plane, with no real component.

Therefore, the real part of

ln(-0.6478) = (ln -1) + (ln 0.6478) = ln 0.6478

Considering what I'd done earlier with your question, including checking the series solution in the "sources" section, I thought I'd done enough. Your characterization of my work is an insult.

The other answer given here -- a good one -- looks more general to me. Instead of being based on the Euler identity that I used here, it's based on Euler's more general formula

e^(ix) = cos x + i sin x

Nevertheless, I felt it necessary to defend what I'd done.

Answer 2

In the context of n = (log 0.05) / (log -0.6478), the numbers
inside the logarithms are complex numbers and not real numbers.

The complex numbers, usually denoted by z is an ordered pair (x,y) in the complex plane which is expressed as:
z = x + y*i,

where x and y are real numbers and i = sqrt(-1). Also note that the first term x corresponds to the real component of z, while the next term, y*i, is the imaginary part of the number z.

As a remark, the set of real numbers (R) is only a subset of the set of complex numbers (C). This happens when the imaginary component of z is 0; that is, y*i = 0 and thus z = x, a real number. So a real number is also a complex number.

Remark: We denote Log (z) as the complex logarithm function evaluated at z.

Now, we go back to the given, we have log (0.05) which can be expressed as Log ( 0.05 + 0i ) and log( -0.6478 ) as Log( -0.6478 + 0i ).

So the given equation above is written as follows:
n = Log ( 0.05 + 0i ) / Log( -0.6478 + 0i ).

Because the expression inside these logarithms are non-zero,
the principal value of the logarithm is defined as:
Log (z) = ln |z| + i*Arg(z)

where |z| = sqrt (x^2 + y^2) and Arg(z) = Arctan(y/x). Note the following:
(1) |z| pertains to the distance (or magnitude) of z from the origin (one can treat z here as a point lying on the x-y plane) and not the usual absolute value of z where the sign of z is omitted;
(2) the ln function here is the usual natural logarithm function we know from Algebra;
(3) Arg(z) is the argument Arg(z) ranges on the closed interval [-pi, pi] and
(4) the values |z|, ln|z| and Arg(z) are real valued.

Hence, we have the following:
Log ( 0.05 + 0i ) = ln |0.05 + 0i| + i * Arg(0.05 + 0i)
=ln ( sqrt( 0.05^2 + 0^2 ) ) + i * Arctan ( 0 / 0.05 )
=ln ( sqrt( 0.025 ) ) + i * Arctan ( 0 )
=ln ( 0.05 ) + i * ( 0 )
=ln ( 0.05 ) and

Log ( -0.6478 + 0i ) = ln |-0.6478 + 0i| + i * Arg(0.6478 + 0i)
=ln ( sqrt( (-0.6478)^2 + 0^2 ) ) + i * Arctan ( 0 / 0.6478 )
=ln ( sqrt( 0.41964484 ) ) + i * Arctan ( 0 )
=ln ( 0.6478 ) + i * ( 0 )
=ln ( 0.6478 ).

In algebra, we know that ln x = log (x) / log e, where
e = 2.7182818284590452353602874713527...

So, from the equation
n = Log ( 0.05 + 0i ) / Log( -0.6478 + 0i )
   = ln ( 0.05 ) / ln ( 0.6478 )
   = [ log ( 0.05 ) / log e ] / [ log ( 0.6478 ) / log e ]
   = log ( 0.05 ) / log ( 0.6478 ) by cancellation of log e
   = -1.3010299956639811952137388947245 / -0.18855905632584188228202786882908
   = 6.8998541943046199651634693712603.


For deeper understanding of the complex numbers and logarithms involving complex numbers, kindly refer to the sites I listed below.