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

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

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

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

Why would you need to use ln?

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.

Is ln the same thing as log base 10?

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

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

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).

What is log_15 (9), if log_15 (5) = a The answer is: 2-2a Could someone explain the steps to solve this?

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!

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

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

Main content

Algebra 2

Course: Algebra 2 > Unit 8

Lesson 4: The change of base formula for logarithms

Logarithm properties review

Google Classroom

Review the logarithm properties and how to apply them to solve problems.

What are the logarithm properties?


Product rule	log, start base, b, end base, left parenthesis, M, N, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, plus, log, start base, b, end base, left parenthesis, N, right parenthesis
Quotient rule	log, start base, b, end base, left parenthesis, start fraction, M, divided by, N, end fraction, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, minus, log, start base, b, end base, left parenthesis, N, right parenthesis
Power rule	log, start base, b, end base, left parenthesis, M, start superscript, p, end superscript, right parenthesis, equals, p, log, start base, b, end base, left parenthesis, M, right parenthesis
Change of base rule	log, start base, b, end base, left parenthesis, M, right parenthesis, equals, start fraction, log, start base, a, end base, left parenthesis, M, right parenthesis, divided by, log, start base, a, end base, left parenthesis, b, right parenthesis, end fraction

Want to learn more about logarithm properties? Check out this video.

Rewriting expressions with the properties

We can use the logarithm properties to rewrite logarithmic expressions in equivalent forms.

For example, we can use the product rule to rewrite log, left parenthesis, 2, x, right parenthesis as log, left parenthesis, 2, right parenthesis, plus, log, left parenthesis, x, right parenthesis. Because the resulting expression is longer, we call this an expansion.

In another example, we can use the change of base rule to rewrite start fraction, natural log, left parenthesis, x, right parenthesis, divided by, natural log, left parenthesis, 2, right parenthesis, end fraction as log, start base, 2, end base, left parenthesis, x, right parenthesis. Because the resulting expression is shorter, we call this a compression.

Problem 1

Current

Expand log, start base, 2, end base, left parenthesis, 3, a, right parenthesis.

Want to try more problems like this? Check out this exercise.

Evaluating logarithms with calculator

Calculators usually only calculate log (which is log base 10) and natural log (which is log base e).

Suppose, for example, we want to evaluate log, start base, 2, end base, left parenthesis, 7, right parenthesis. We can use the change of base rule to rewrite that logarithm as start fraction, natural log, left parenthesis, 7, right parenthesis, divided by, natural log, left parenthesis, 2, right parenthesis, end fraction and then evaluate in the calculator:

\begin{aligned} \log_2(7)&=\dfrac{\ln(7)}{\ln(2)} \\\\ &\approx 2.807 \end{aligned}

Problem 1

Current

Evaluate log, start base, 3, end base, left parenthesis, 20, right parenthesis.
Round your answer to the nearest thousandth.

Want to try more problems like this? Check out this exercise.

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Joey Jang
Posted 6 years ago. Direct link to Joey Jang's post “How did a mathematician f...”
How did a mathematician find e? What's its origin?
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(18 votes)
Answer
- Divy Shah
  Posted 3 years ago. Direct link to Divy Shah's post “It is often called Euler'...”
  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
  Button navigates to signup page
  (6 votes)
Matthew Johnson
Posted 6 years ago. Direct link to Matthew Johnson's post “in `log_1(1)=x`, doesn't ...”
in log_1(1)=x, doesn't x = infinity?
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(8 votes)
Answer
- Kim Seidel
  Posted 6 years ago. Direct link to Kim Seidel's post “See the "Restrictions" se...”
  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
  Button navigates to signup page
  (15 votes)
svlohit2012
Posted 4 years ago. Direct link to svlohit2012's post “Why would you need to use...”
Why would you need to use ln?
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(7 votes)
Answer
- SherlockHolmes.42
  Posted 4 years ago. Direct link to SherlockHolmes.42's post “The natural log function,...”
  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.
  Comment on SherlockHolmes.42's post “The natural log function,...”
  (7 votes)
brandon_bolster
Posted 5 years ago. Direct link to brandon_bolster's post “Is ln the same thing as l...”
Is ln the same thing as log base 10?
Button navigates to signup pageComment on brandon_bolster's post “Is ln the same thing as l...”
(5 votes)
Answer
- Oliver
  Posted 5 years ago. Direct link to Oliver's post “The "log" key on a calcul...”
  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
  Button navigates to signup page
  (6 votes)
Haoyu Wang
Posted a year ago. Direct link to Haoyu Wang's post “why e^𝝿i+1=0? How did Eu...”
why e^𝝿i+1=0?
How did Euler proof this equation?
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(5 votes)
Answer
- KLaudano
  Posted a year ago. Direct link to KLaudano's post “It is a specific case of ...”
  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).
  Button navigates to signup page
  (3 votes)
Muhammadaminbek
Posted 6 years ago. Direct link to Muhammadaminbek's post “Can anyone explain to me ...”
Can anyone explain to me how to solve e^ln^2 x +x^lnx =2e^4
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(5 votes)
Answer
Sellov
Posted 3 years ago. Direct link to Sellov's post “What is log_15 (9), if lo...”
What is log_15 (9), if log_15 (5) = a
The answer is: 2-2a

Could someone explain the steps to solve this?
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(4 votes)
Answer
- Katriana
  Posted 2 years ago. Direct link to Katriana's post “I can explain how to chec...”
  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!
  Button navigates to signup page
  (3 votes)
Matthew Johnson
Posted 6 years ago. Direct link to Matthew Johnson's post “Why is the base 10 logari...”
Why is the base 10 logarithmic scale the standard for calculators?
Button navigates to signup pageComment on Matthew Johnson's post “Why is the base 10 logari...”
(1 vote)
Answer
- Judith Gibson
  Posted 6 years ago. Direct link to Judith Gibson's post “Probably because the rest...”
  Probably because the rest of our number system is built around powers of 10 --- tens, hundreds, thousands, etc. and tenths, hundredths, thousandths, etc.
  Comment on Judith Gibson's post “Probably because the rest...”
  (7 votes)
Andy Huang
Posted 5 years ago. Direct link to Andy Huang's post “How do you do log base 2 ...”
How do you do log base 2 x + log base 3 x = 4?
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(4 votes)
Answer
Julicz
Posted 2 years ago. Direct link to Julicz's post “e is the base of the Natu...”
e is the base of the Natural Logarithms (invented by John Napier), how did he did it and did he calculated the logarithms by hand?
Button navigates to signup pageComment on Julicz's post “e is the base of the Natu...”
(2 votes)
Answer