Log function restrictions - why the base cannot be negative

[@under_the_night_sky5380]

Thank you so much! This is exactly the explanation that I was looking for.

[@BraverMath]

Glad it helped! I guess one of my students had the same question you did :)

[@aseelmuhammed8788]

Yeh ... So helpful ...

[@therovingmaverick4133]

Can't see anything

[@mejercit]

Basically, it is not a function if the base is negative.

[@BraverMath]

Yes. The short answer is, "things" happen that we don't like....and "things" we do like don't happen haha.

[@TheProGamerMC20]

@@BraverMathyou mean this? log₋₂(-8) = ln(-8) / ln(-2) = ln(-1 × 8) / ln(-1 × 2) = ln(-1) + ln(8) / ln(-1) + ln(8) also, ln(8) = ln(2³) = 3ln(2) let -1 = e^iπ = ln(e^iπ) + ln(8) / ln(e^iπ) + ln(8) = πln(e)i + ln(8) / πln(e)i + ln(8) multiply by configure, foil, factor if necessary, etc. ?

[@lindazhou7906]

Thank you so much! Now log functions make so much more sense!

[@BraverMath]

If it helped even 1 person, my job is done. Glad it was useful to you in your journey.

[@thabisochokocha5347]

​​@@BraverMath🤝

[@elite7585]

omg thank you !!

[@Hala-rv2ic]

thank u so much

[@ritira20mila]

Yes, so if "the logarithm is the exponent to be given to the base (base other than 1) to get the argument", THE BASE CAN BE NEGATIVE. It's just that if we require the logarithm to define an continuous function, the base must be positive. *But who says that we want a function? And who says that we want that function to be continuous? I am perfectly happy with a logarithm without its function, or with a discontinuous function that includes negative bases*. The problem is when they teach you that "the logarithm is the exponent to be given to the base (base other than 1) to get the argument". Flat WRONG. If that was true, some negative bases would give you as logarithms some absolutely respectable REAL numbers. No need to go into complex numbers. (Examples are in the video). But the actual definition is different. They define this thing called logarithm with a special requirement: a positive base (and therefore a positive argument). But that definition is not equal to "the exponent to be given to the base (base other than 1) to get the argument". Teachers and books that tell you that the logarithm is "the exponent to be given to the base (base other than 1) to get the argument" are WRONG. What they mean is "the logarithm is the exponent to be given to the base (base other than 1) to get the argument, but only with a positive base, by convention, not with any mathematical basis, but for our whimsical disfavor of discontinuos functions".

[@mejercit]

Base 1 can be defined.

[@vaibhav2831]

Thank sir

[@justabunga1]

For logarithmic function, it should be the base must be positive other than 1. The argument must always be positive. This is similar like exponential functions where the base must always be positive not equal to 1.