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Mathematical Expression Editor
Motivating Questions
How is the base- logarithm defined?
What is the “natural logarithm” and how is it different from the base-
logarithm?
How can we solve an equation that involves to some unknown quantity?
In previous sections, we introduced the idea of an inverse function. The
fundamental idea is that has an inverse function if and only if there exists another
function such that and “undo” one another’s respective processes. In other
words, the process of the function is reversible to generate a related function
.
More formally, recall that a function (where goes from a set to a set ) has an
inverse function if and only if there exists another function going from to such that
for every in the domain of and for every in the range of . We know that given a
function , we can use the Horizontal Line Test to determine whether or not has an
inverse function. Finally, whenever a function has an inverse function, we call its
inverse function and know that the two equations and say the same thing from
different perspectives.
Let be the “powers of 10” function, which is given by .
a.
Complete the following table to generate certain values of .
b.
Why does have an inverse function?
c.
Since has an inverse function, we know there exists some other function,
say , such that writing “” says the exact same thing as writing “”. In
words, where produces the result of raising to a given power, the function
reverses this process and instead tells us the power to which we need to
raise , given a desired result. Complete the table to generate a collection
of values of .
d.
What are the domain and range of the function ? What are the domain
and range of the function ?
The base- logarithm
The powers-of- function is an exponential function with base . As such, is always
increasing, and thus its graph passes the Horizontal Line Test, so has an inverse
function. We therefore know there exists some other function, , such that writing is
equivalent to writing . For instance, we know that and , so it’s equivalent to say that
and . This new function we call the base logarithm, which is formally defined as
follows.
Given a positive real number , the base- logarithm of is the power to which
we raise to get . We use the notation “” to denote the base- logarithm of
.
The base- logarithm is therefore the inverse of the powers of function. Whereas
takes an input whose value is an exponent and produces the result of taking to that
power, the base- logarithm takes an input number we view as a power of and
produces the corresponding exponent such that to that exponent is the input
number.
In the notation of logarithms, we can now update our earlier observations with the functions
and and see how exponential equations can be written in two equivalent ways. For
instance,
each say the same thing from two different perspectives. The first says is to
the power , while the second says is the power to which we raise to get .
Similarly,
If we rearrange the statements of the facts, we can see yet another important
relationship between the powers of and base- logarithm function. Noting that and
are equivalent statements, and substituting the former equation into the latter shows,
we see that
In words, the equation says that “the power to which we raise to get , is ”.
That is, the base- logarithm function undoes the work of the powers of
function.
In a similar way, we can observe that by replacing with we
have
In words, this says that “when is raised to the power to which we raise in order to
get , we get ”.
We summarize the key relationships between the powers-of- function and its
inverse, the base- logarithm function, more generally as follows. Let and .
The domain of is the set of all real numbers and the range of is the set
of all positive real numbers.
The domain of is the set of all positive real numbers and the range of is
the set of all real numbers.
For any real number , . That is, .
For any positive real number , . That is, .
and .
The base- logarithm function is like the sine or cosine function in this way: for certain
special values, it’s easy to know by heart the value of the logarithm function. While
for sine and cosine the familiar points come from specially placed points on the unit
circle, for the base- logarithm function, the familiar points come from powers of . In
addition, like sine and cosine, for all other input values, (a) calculus ultimately
determines the value of the base- logarithm function at other values, and (b) we
use computational technology in order to compute these values. For most
computational devices, the command produces the result of the base- logarithm of
.
It’s important to note that the logarithm function produces exact values. For
instance, if we want to solve the equation , then it follows that is the exact solution
to the equation. Like or , is a number that is an exact value. A computational
device can give us a decimal approximation, and we normally want to distinguish
between the exact value and the approximate one. For the three different numbers
here, , , and .
For each of the following equations, determine the exact value of the unknown
variable. If the exact value involves a logarithm, use a computational device to also
report an approximate value. For instance, if the exact value is , you can also note
that .
a.
b.
c.
d.
e.
f.
g.
The natural logarithm
The base- logarithm is a good starting point for understanding how logarithmic
functions work because powers of are easy to mentally compute. We could similarly
consider the powers of or powers of function and develop a corresponding
logarithm of base or . But rather than have a whole collection of different
logarithm functions, in the same way that we now use the function and
appropriate scaling to represent any exponential function, we develop a single
logarithm function that we can use to represent any other logarithmic function
through scaling. In correspondence with the natural exponential function, , we
now develop its inverse function, and call this inverse function the natural
logarithm.
Given a positive real number , the natural logarithm of is the power to which we
raise to get . We use the notation “” to denote the natural logarithm of
.
We can think of the natural logarithm, , as the “base- logarithm”. For
instance,
and
The former equation is true because “the power to which we raise to get is ”; the
latter equation is true since “when we raise to the power to which we raise to get ,
we get ”.
Let and be the natural exponential function and the natural logarithm
function, respectively.
a.
What are the domain and range of ?
b.
What are the domain and range of ?
c.
What can you say about for every real number ?
d.
What can you say about for every positive real number ?
e.
Complete the following tables with both exact and approximate of and .
Then, plot the corresponding ordered pairs from each table on the axes
below and connect the points in an intuitive way. When you plot the
ordered pairs on the axes, in both cases view the first line of the table as
generating values on the horizontal axis and the second line of the table as
producing values on the vertical axis label each ordered pair you plot
appropriately.
or logarithms in general
In the previous sections, we looked at two specific (and the most common) types of
logarithms, base-10 and natural log. In order to fully discuss logarithms, we need to
talk about logarithms in general with any base. Let . Because the function has an
inverse function, it makes sense to define its inverse like we did when or . The base-
logarithm, denoted is defined to be the power to which we raise to get
.
Evaluate the following base- logarithms.
(a)
(b)
(a)
(b)
Revisiting
In earlier sections, we saw that that function plays a key role in modeling
exponential growth and decay, and that the value of not only determines whether
the function models growth () or decay (), but also how fast the growth or decay
occurs. Furthermore, once we introduced the natural base , we realized that we could
write every exponential function of form as a horizontal scaling of the function by
writing
for some value . Our development of the natural logarithm function in the current
section enables us to now determine exactly.
Determine the exact value of for which .
Since we want to hold for every value of and , we need to have , and thus .
Therefore, is the power to which we raise to get , which by definition means that
.
In modeling important phenomena using exponential functions, we will frequently
encounter equations where the variable is in the exponent, like in the example where
we had to solve . It is in this context where logarithms find one of their most
powerful applications.
Solve each of the following equations for the exact value of
the unknown variable. If there is no solution to the equation, explain why
not.
a.
b.
c.
d.
e.
f.
g.
h.
a.
b.
c.
d.
e.
f.
g.
No solution, because is outside of the range of
h.
(a)
The base- logarithm of , denoted is defined to be the power to which we
raise to get . For instance, , since . The function is thus the inverse of
the powers-of- function, .
(b)
The natural logarithm differs from the base- logarithm in that it is the
logarithm with base instead of , and thus is the power to which we raise
to get . The function is the inverse of the natural exponential function .
(c)
The natural logarithm often enables us solve an equation that involves to some
unknown quantity. For instance, to solve , we can first solve for by subtracting
from each side and dividing by to get
This last equation says “ to some power is ”. We know that it is equivalent to
say
Since is a number, we can solve this most recent linear equation for . In
particular, , so