My teacher told me that the natural logarithm of a negative number does not exist, but $$\ln (-1)=\ln (e^ {i\pi})=i\pi$$ So, is it logical to have the natural logarithm of a negative number?
The point is: the complex logarithm is not a function, but what we call a multivalued function. To turn it into a proper function, we must restrict what $\theta$ is allowed to be, for example $\theta \in (-\pi,\pi]$. This is called the principal complex logarithm and is usually denoted by $\operatorname {Log}$ (capital L).
Logarithms are defined as the solutions to exponential equations and so are practically useful in any situation where one needs to solve such equations (such as finding how long it will take for a population to double or for a bank balance to reach a given value with compound interest). Historically, they were also useful because of the fact that the logarithm of a product is the sum of the ...
You can define everything you want, but will this newborn object satisfy properties you want, depends on your definition. Assume, we do have logarithms for negative numbers and zero and all the properties of logarithms are preserved. Then we immediately obtain a contradiction. Here it is $$ 0=\log 1=\log (-1)^2=2\log (-1) $$ so $\log (-1)=0$ and from the definition of logarithms we have $-1=10 ...
Shortly after the work of Napier, Briggs, inspired by that work, produced tables of the base $10$ logarithm. Related tables were used for computations for centuries.
The discrete Logarithm is just reversing this question, just like we did with real numbers - but this time, with objects that aren't necessarily numbers. For example, if $ {a\cdot a = a^2 = b}$, then we can say for example $ {\log_ {a} (b)=2}$.
Thank you for the answer. I am aware of the general solutions for complex numbers. In my question above I am specifically asking to the definition for real numbers. It is in that scenario that I have always only understood logarithms as defined for positive numbers, although there seems to be solutions for negative bases. My apologies if that wasn't clear.
Currently, in my math class, we are learning about logarithms. I understand that the common logarithm has a base of 10 and the natural has a base of e. But, when do we use them? For example the equ...
I would like to know how logarithms are calculated by computers. The GNU C library, for example, uses a call to the fyl2x() assembler instruction, which means that logarithms are calculated directl...
Here I was exposed to so many variations: Saying the two letters l n Saying "log"/"logarithm" Saying "natural log" Saying "log e" All of the above were native-English speakers from different parts of the world. No one pronounced it like we Israelis do, as "lan". As for your "linn", I believe it was a New Zealander. Their e's sound like i's ...
