What to do if the number after log expression is negative?

Question

example: log (-10), i faced this problem when i calculated Gain...

just for more information im caluculating the gain of inverting amplifier, where (Voutput/ Vinput) was negative number.

Answer 1

The answer would be a complex number.

log(-a)=log(-1*a)
=log(-1)+log(a)

-1=e^(i*pi)
log(-1)=log(e^(i*pi))
=i*pi log(e)

log(-a)=i*pi*log(e)+log(a)
So, log(-10)=i*pi*log(e)+1

However, instead of saying -1=e(i*pi), you can also say -1=e^(i*3pi) or 5pi 7 pi..., all of which would slightly change the final answer. Therefore, the log of a negative number has an infinite number of values.

Answer 2

Let's take a logarithm of base b of a number x, which I will denote as simply denote as log(x). Remember that the base can be any positive number and x can be any real number (negative, zero, positive).

If log(x) = N, then b^N = x.

Since b is positive, there is no N that will yield a negative x.

But don't take my word for it, let's give it a try:

(3)^3 = 27 > 0

(3)^0 = 1 > 0

(3)^-3 = 1/(3)^3 = 1/27 > 0

(10)^-20 = 0.00000000000000000001 > 0

Go ahead and try some more. Let me know when you find one. Good luck! :)

Answer 3

a logarithm is the power to which
a stated number called the base is
raised to yield a specific number-
that is,
y=e^x and is always +ve
{x is the log and e is the base}
take ln of each side
x=lny
since y is always +ve,y has to
be +ve for x to be in the real
number domain-if y is -ve,
x is a complex number
-10 can be written
10(cos(t)+i*sin(t))
where t=pi radians

Log(z)=ln(mod(z))
+i*Arg(z)
{z member C -{0}}
hence,
Log(-10)
=ln(10)+i*pi

[note;Log(-10) is not equal
to ln(1/10),
-ln(10)= ln(1/10),
ln(0) is undefined]

i hope that this helps

Answer 4

Negative numbers don't have log (at least in real numbers) Certainly you made a mistake

Answer 5

Note that - 1 = 1 at angle π = e^(i π)
So log ( - 1) = i π

log (- 10 ) = log(10 x - 1)
= log 10 + log( - 1)
= log 10 + i.π

Answer 6

like someone else said, the answer would be complex for the reason he said.

however, given that you said you were calculating "Gain" i suspect that you've made an error earlier in your calculation. Gain, whether optical or electrical is a physical measurable meaning that it can never be complex.

Answer 7

u dont really need the negative sign in the gain.... cuz its just to show that the op-amp is an inverted one.

Answer 8

if the no. after log is negative then u hve to invert the no.
like log(-x)


log(1/x)

Answer 9

that is not possible.
it's a math error.
domain of a log function does not allow non-positive numbers.