- #1
rudders93
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Homework Statement
Hi, I was wondering why: e^ln(2) = 2. I'd just like to see how to get this result instead of just having to memorize it
Thanks!
Why Does e Raised to the Natural Log of 2 Equal 2?
- Thread starter rudders93
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In summary, the homework statement is asking why e^ln(2) = 2. The person explains that ln(y) is the power you need to raise e to get y, and that e^ln(y)=y. This can be simply verified by the Power Rule of Exponents.
Hi, I was wondering why: e^ln(2) = 2. I'd just like to see how to get this result instead of just having to memorize it
Thanks!
- #2
- 9,568
- 775
Take the natural logarithm of both sides and remember that two positive numbers whose logs are equal must be equal.
- #3
Tobias Funke
- 132
- 40
It's simply a matter of definition, nothing more. You need to fully understand inverse functions and the conditions under which a function has an inverse. e^x takes any number and outputs a positive real number (I'm assuming you're restricting your attention to real numbers) and it's also one to one and onto the positive reals, so it has an inverse function that takes any positive number and returns some real number.
So you have some number x and you exponentiate to get e^x. Now, every positive number y is the result of e raised to some power (it's not a proof, but looking at the graph of e^x will probably convince you of this). This power is called ln(y).
A verbal description would be that ln(y) is the power you need to raise e to to get y. Then e^ln(y) is e raised to the power you need to raise e to to get y. In other words, e^ln(y)=y. When it's stated that way, it's not so mysterious.
So you have some number x and you exponentiate to get e^x. Now, every positive number y is the result of e raised to some power (it's not a proof, but looking at the graph of e^x will probably convince you of this). This power is called ln(y).
A verbal description would be that ln(y) is the power you need to raise e to to get y. Then e^ln(y) is e raised to the power you need to raise e to to get y. In other words, e^ln(y)=y. When it's stated that way, it's not so mysterious.
- #4
rudders93
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Ah ok. Thanks!
- #5
annoymage
- 362
- 0
e^ln(2) = x
and solve for x.
and solve for x.
- #6
paulfr
- 193
- 3
There is a much deeper and more fundamental reason that
e^ln(2) = 2
Exponentiation and Logarithms are INVERSE FUNCTIONS
Briefly, inverse functions, in layman's terms, are functions which when performed
in succession return you to where you started.
For example; start with x, add 3 to it and then subtract three from it and you are back at x
Addition and Subtraction are Inverse Functions.
Other inverse function pairs are;
1/ Multiplication and Division
2/ Powers and Roots
3/ Trig and Inverse Trig functions
4/ Integration and Differentiation {This is the Fundamental Theorem of Calculus}
Note the reverse of the original expression is ... Ln e^2 is also = 2
This can be simply verified by the Power Rule of Exponents
Ln e^2 = 2 Ln e = 2 x 1 = 2
An important result of this is that whenever you need to solve an
equation, the operation most likely to get you quickly to your answer
is to perform the Inverse Function of the outer operation to both sides.
e^ln(2) = 2
Exponentiation and Logarithms are INVERSE FUNCTIONS
Briefly, inverse functions, in layman's terms, are functions which when performed
in succession return you to where you started.
For example; start with x, add 3 to it and then subtract three from it and you are back at x
Addition and Subtraction are Inverse Functions.
Other inverse function pairs are;
1/ Multiplication and Division
2/ Powers and Roots
3/ Trig and Inverse Trig functions
4/ Integration and Differentiation {This is the Fundamental Theorem of Calculus}
Note the reverse of the original expression is ... Ln e^2 is also = 2
This can be simply verified by the Power Rule of Exponents
Ln e^2 = 2 Ln e = 2 x 1 = 2
An important result of this is that whenever you need to solve an
equation, the operation most likely to get you quickly to your answer
is to perform the Inverse Function of the outer operation to both sides.
- #7
log10power
- 1
- 0
e^ln2 = 2
Apply ln to both sides: ln e^ln2 = ln 2
ln2 ln e = ln 2
Since ln e = 1, so ln2 (1) = ln 2
ln2 = ln2
Apply ln to both sides: ln e^ln2 = ln 2
ln2 ln e = ln 2
Since ln e = 1, so ln2 (1) = ln 2
ln2 = ln2
FAQ: Why Does e Raised to the Natural Log of 2 Equal 2?
What is the proof that e^ln(2) is equal to 2?
The proof is based on the properties of logarithms and exponentials. Since ln(2) is the natural logarithm of 2, it can be rewritten as loge(2). Using the property that e^ln(x) = x, we can rewrite e^ln(2) as e^loge(2). This simplifies to just 2, proving that e^ln(2) = 2.
Why is the natural logarithm used in this proof?
The natural logarithm, denoted as ln(x), is the inverse of the natural exponential function, e^x. This means that ln(x) can "undo" the effect of e^x. In this proof, we are trying to show that e^ln(2) equals 2, so it makes sense to use the inverse function of e^x, which is ln(x).
What is the significance of e in this proof?
The number e, also known as Euler's number, is a mathematical constant that is approximately equal to 2.71828. It is an important number in the study of exponential functions and is often used in mathematical proofs involving logarithms and exponentials.
Can this proof be applied to other numbers besides 2?
Yes, this proof can be applied to any number. In fact, the general proof is e^ln(x) = x, where x is any positive real number. This means that e^ln(2) = 2 is just a specific case of this general proof.
Is this proof a fundamental concept in mathematics?
Yes, this proof is a fundamental concept in mathematics that is used in many different areas, including calculus, algebra, and number theory. It is also a key part of understanding the relationship between logarithms and exponentials.