The Number e
is an important mathematical constant, approximately equal to
. When used as the base for a logarithm, we call that logarithm the natural logarithm and write it as
Recognize the properties and uses of the number
- The natural logarithm, written , is the power to which must be raised to obtain .
- The constant can be defined in many ways, most of which involve calculus. For example, it is the limit of the sequence whose general term is . Also, it is the unique number so that the area under the curve from to is square unit.
- e: The base of the natural logarithm, 2.718281828459045…
- logarithm: The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.
, sometimes called the natural number, or Euler's number, is an important mathematical constant approximately equal to 2.71828. When used as the base for a logarithm, the corresponding logarithm is called the natural logarithm, and is written as
. Note that
There are a number of different definitions of the number
. Most of them involve calculus. One is that
is the limit of the sequence whose general term is
. Another is that
is the unique number so that the area under the curve
Another definition of
involves the infinite series
. It can be shown that the sum of this series is
is very important in mathematics, alongside
All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in the formulation of Euler's identity, which (amazingly) states that
Like the constant
is irrational (it cannot be written as a ratio of integers), and it is transcendental (it is not a root of any non-zero polynomial with rational coefficients).
One of the many places the number
plays a role in mathematics is in the formula for compound interest. Jacob Bernoulli discovered this constant by asking questions related to the amount of money in an account after a certain number of years, if the interest is compounded
times per year. He was able to come up with the formula that if the interest rate is
percent and is calculated
times per year, and the account originally contained
dollars, then the amount in the account after
years is given by
By then asking about what happens as
gets arbitrarily large, he was able to come up with the formula for continuously compounded interest, which is
Graphs of Exponential Functions, Base e
is a basic exponential function with some very interesting properties.
Identify important properties about the graph of
- The function is a function which is very important in calculus. It appears in many applications.
- The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value.
- The graph of lies between the graphs of and .
- tangent: A straight line touching a curve at a single point without crossing it.
- exponential function: Any function in which an independent variable is in the form of an exponent; they are the inverse functions of logarithms.
- asymptote: A line that a curve approaches arbitrarily closely, as it extends toward infinity.
The basic exponential function, sometimes referred to as the exponential function, is
is the number (approximately 2.718281828) described previously. Its graph lies between the graphs of
. The graph's
-intercept is the point
, and it also contains the point
Sometimes it is written as
The graphs of , , and :
The graph of
lies between that of
The graph of
is upward-sloping, and increases faster as
increases. The graph always lies above the
-axis, but gets arbitrarily close to it for negative
; thus, the
-axis is a horizontal asymptote. The graph of
has the property that the slope of the tangent line to the graph at each point is equal to its
-coordinate at that point.
is the only function with this property.
A Model For Proportional Change
The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i. e., percentage increase or decrease) in the dependent variable. If the change is positive, this is called exponential growth and if it is negative, it is called exponential decay. For example, because a radioactive substance decays at a rate proportional to the amount of the substance present, the amount of the substance present at a given time can be modeled with an exponential function. Also, because the the growth rate of a population of bacteria in a petri dish is proportional to its size, the number of bacteria in the dish at a given time can be modeled by an exponential function such as
is the number of bacteria present initially (at time
is a constant called the growth constant.
The natural logarithm is the logarithm to the base e
, where e
is an irrational and transcendental constant approximately equal to 2.718281828.
Identify some properties and uses of the natural logarithm
- The natural logarithm is the logarithm with base equal to e.
- The number e and the natural logarithm have many applications in calculus, number theory, differential equations, complex numbers, compound interest, and more.
- natural logarithm: The logarithm in base e.
- e: The base of the natural logarithm, approximately 2.718281828459045…
The Natural Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. The natural logarithm is the logarithm with base equal to e
The natural logarithm can be written as
but is usually written as
. The two letters l and n are reversed from the order in English because it arises from the French (logarithm naturalle).
Just as the exponential function with base
arises naturally in many calculus contexts, the natural logarithm, which is the inverse function of the exponential with base
, also arises in naturally in many contexts. It is used much more frequently in physics, chemistry, and higher mathematics than other logarithmic functions. For example, the doubling time for a population which is growing exponentially is usually given as
is the growth rate, and the half-life of a radioactive substance is usually given as
is the decay constant.
The function slowly grows to positive infinity as
increases and rapidly goes to negative infinity as
("slowly" and "rapidly" as compared to any power law of
-axis is an asymptote. The graph of the natural logarithm lies between that of
. Its value at
, while its value at
The graphs of , , and :
The graph of the natural logarithm lies between the base 2 and the base 3 logarithms.
Solving Equations Using
The natural logarithm function can be used to solve equations in which the variable is in an exponent.
Example: Find the positive root of the equation
The first step is to take the natural logarithm of both sides:
Using the power rule of logarithms it can then be written as:
Dividing both sides by
Thus the positive solution is
This can be calculated (approximately) with any scientific handheld calculator.
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