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Natural Exponential Function and Natural Logarithmic Function

Topical Outline | Algebra 2 Outline | MathBits' Teacher Resources

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In this lesson, we will begin our work with the number e.

e is an irrational number, approximately 2.71828183,
named after the 18th century Swiss mathematician, Leonhard Euler.
The number e is one of the famous numbers in mathematics. As your mathematical studies progress from Algebra 2 through Calculus, you will discover that the role of e in mathematics is equal in importance to
that of the number π.

π is the ratio between circumference and diameter shared by all circles.
e is the base rate of growth shared by all continually growing processes.


There are 5 numbers that are considered the "five most important numbers in mathematics". The five numbers are 0, 1, π, e, and i.    Now, you know them all!


Natural Exponential Function
When the base, b, of the exponential function y = bx, is replaced with e,
we have the
natural exponential function.
The natural exponential function may be expressed as y = ex or as y = exp(x).
In functional notation: f (x) = ex or f (x) = exp(x)

The graph of the function defined by f (x) = ex
looks similar to the graph of f (x) = bx where b > 1.

This natural exponential function is simply a "version" of the exponential function f (x) = bx

As such, the characteristics of this graph are similar to the characteristics of the exponential graph.

Domain: All Reals
Range: y > 0

The function f (x) = ex is the only function where the slope of a tangent to the curve at any point is equal to the height of the curve at that point.
This fact will look
"more exciting"
when you get to Calculus.


Inverse of f (x) = ex
Since f (x) = ex is a one-to-one function, we
know that its inverse will also be a function.
But what is the equation of the inverse of f (x) = ex ?

The equation of the inverse is:
y = loge x = ln x
and is called the natural logarithmic function.

This new function is simply a "version" of
y = logb x where b > 1.

On the Graph: Notice how (0,1) from f(x) = ex becomes (1,0) for y = ln(x).
The coordinates switch places between a graph and its inverse.


Natural Logarithmic Function
When the base, b, of the logarithmic function, y = logb x, is replaced with e,
we have the
natural logarithmic function.

The natural logarithmic function, y = loge x, is more commonly written y = ln x.
In functional notation: f (x) = ln x

The graph of the function defined by y = ln x,
looks similar to the graph of y = logb x where b > 1.

The characteristics of this new function are similar to logarithmic function characteristics we already know.

Domain: x > 0
Range: All Reals



For help with exponential expressions on your calculator, click here.


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