Question 1

How did a mathematician find e? What's its origin?

Accepted Answer

It is often called Euler's number after Leonhard Euler (pronounced "Oiler")
e is an irrational number (it cannot be written as a simple fraction).

e is the base of the Natural Logarithms (invented by John Napier).

e is found in many interesting areas, so is worth learning about. you can check this link to find out:
https://www.khanacademy.org/math/in-in-grade-11-ncert/in-in-exponential-and-logarithmic-functions/copy-of-math3-e-and-natural-log/v/e-through-compound-interest and https://www.khanacademy.org/math/in-in-grade-11-ncert/in-in-exponential-and-logarithmic-functions/copy-of-math3-e-and-natural-log/v/e-as-limit

Question 2

in `log_1(1)=x`, doesn't `x = infinity`?

Accepted Answer

See the "Restrictions" section at this link (about 1/2 down page).  The base is restricted from being 1.
https://www.khanacademy.org/math/algebra2/exponential-and-logarithmic-functions/introduction-to-logarithms/a/intro-to-logarithms

Question 3

Why would you need to use ln?

Accepted Answer

The natural log function, ln, is the log with a base of Euler's number, e.

Here is an example of when it can be used:
e^x = 2
--> To solve for x, we would take the ln of both sides. This is because x is the exponent of e, and the e and natural log will cancel out when put together.
ln(e^x) = ln(2)
x = ln(2)

This is the most common way I've seen the natural log used, but there are no doubt other ways to use it.

Question 4

why e^𝝿i+1=0?
How did Euler proof this equation?

Accepted Answer

It is a specific case of his formula, e^(i*x) = cos(x) + i*sin(x). The proof of this formula requires Calculus level math though (e.g. power series or differentiation).

Question 5

Is ln the same thing as log base 10?

Accepted Answer

The "log" key on a calculator is log base 10. "ln" is natural logarithm, and there is a video for that here: https://www.khanacademy.org/math/algebra2/exponential-and-logarithmic-functions/e-and-the-natural-logarithm/v/natural-logarithm-with-a-calculator

Question 6

What is log_15 (9), if log_15 (5) = a
The answer is: 2-2a

Could someone explain the steps to solve this?

Accepted Answer

I can explain how to check it, at least, though I'm not sure that this is how you would originally come to this answer.
log_15(5)=a, and we want to see whether log_15(9)=2-2a. We can begin by finding a.
log_15(5)=a
log(5)/log(15)=a Use the change of base rule.
a=approx. 0.5943 
Next we can find 2-2a:
2-2*approx.0.5943 Try to use the entire answer you got for a, instead of a rounded one, if you can.
2-approx. 1.1886=approx. 0.8114
Now we can find log_15(9), and see if it equals 0.8114.
log_15(9)=log(9)/log(15) Change of base rule.
log(9)/log(15)=approx. 0.8114
log_15(9)=2-2a True.
This shows that it is indeed the case that if log_15(5)=a, then log_15(9)=2-2a, but it does not seem like it is the way that you would come to figure it in the first place.
Perhaps if you figured that a=approx. 0.5943, and that log_15(9)=approx. 0.8114, you might just happen to notice that 0.8114+2(0.5943)=approx. 2. I have not figured anything better than this for this question. Maybe someone else will. For now, I hope you have a good day. Keep going!

Question 7

Why is the base 10 logarithmic scale the standard for calculators?

Accepted Answer

Probably because the rest of our number system is built around powers of 10 --- tens, hundreds, thousands, etc.   and tenths, hundredths, thousandths, etc.


Product rule	$\log_{b} (M N) = \log_{b} (M) + \log_{b} (N)$ ‍
Quotient rule	$\log_{b} (\frac{M}{N}) = \log_{b} (M) - \log_{b} (N)$ ‍
Power rule	$\log_{b} (M^{p}) = p \log_{b} (M)$ ‍
Change of base rule	$\log_{b} (M) = \frac{\log_{a} (M)}{\log_{a} (b)}$ ‍

Course: Algebra (all content) > Unit 11

Logarithm properties review

What are the logarithm properties?

Rewriting expressions with the properties

Evaluating logarithms with calculator

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