The base 10 logarithm function
Background: Every positive number, y,
can be expressed as 10 raised to some
power, x. This relationship is described by the equation
y = 10 x,
and described by this graph:
For example the number 16 can be expressed as 10 1.2.
This is the black dot in the graph. We define a function called the base 10 logarithm that takes a number like 16
as input, calculates that it can be expressed as 10 1.2,
and returns the exponent 1.2 as its output value:
Here is the formal definition of the base 10 logarithm function.
Definition: The base 10 logarithm is the function that takes
any positive number x as input and returns the
exponent to which the base 10 must be raised to obtain x.
It is denoted log(x).
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Example 1: Evaluate log ( 1000 ).
The argument of the
logarithm function (i.e. the quantity in brackets, 1000), is easily expressed as 10
raised to the exponent 3. The logarithm function then returns the exponent.
log ( 1000 ) = log ( 10 3 ) = 3
Example 2: Evaluate log ( 10 5.7 ).
The argument is already expressed
as 10 raised to an exponent, so the logarithm function simply returns the exponent.
log ( 10 5.7 ) = 5.7
Example 3: Evaluate log ( 16 ).
According to the graph 16 = 10 1.2.
The logarithm function returns the exponent 1.2.
log ( 16 ) = log ( 10 1.2 ) = 1.2
Graph: The blue curve shown to the right is the graph of the base 10 logarithm function, y = log(x).
Notice that for any positive x it is single valued and for any negative x it is undefined.
If you compare this graph to the graph
of y = 10 x above
then you see that one can be gotten from the other by interchanging
the x and y axes.
For comparison the red curve is the graph of the natural logarithm function (y = ln(x), covered in the next section). The natural logarithm graph has exactly the same shape
as the base 10 logarithm graph; it is just 2.3 times as tall.
An important feature of logarithm functions (no matter what base) is that they increase very slowly
as x becomes very large. They describe nicely how the human
ear perceives loudness and the way the human eye perceives brightness.
Domain and range:
The domain of the base 10 logarithm function is all positive real numbers and the range is
all real numbers.
The base 10 logarithm function can be
extended to the complex numbers,
in which case the domain is all complex numbers except zero.
The base 10 logarithm of zero is always undefined.
Some special values of the base 10 logarithm function
Solving the equation 10 x = c for x
by using the base 10 logarithm function
Suppose that x is unknown but that 10 x equals
a known value c. Then finding x requires solving the
following equation for x.
10 x = c
The solution is
x = log (c)
This is because finding log (c) means expressing
c as 10 to an exponent, and then returning that exponent.
But the original equation says that that exponent is x.
Note that if c is negative then there is no real solution.
However there is a complex solution.
Furthermore if c = 0 then there is no solution at all.
log (x) and 10 x are inverse functions
Consider the 10 x function which takes x and returns
10 x, like this:
The base 10 logarithm function is defined to do exactly the opposite, namely:
Therefore these are inverse functions.
Note the following:
- Because the 10 x function is the inverse of
the base 10 logarithm function it is sometimes called the antilogarithm function.
- We saw above that the solution of
10 x = y is
x = log (y). We should look at these two equations
as expressing the same relationship between x and y but
from different points of view. The first equation is the relationship solved for
y and the second one is the relationship solved for x.
(An analogy is that the statement “Tom is Jane’s brother”
is equivalent to the statement that “Jane is Tom’s sister”.)
- In the previous bullet we saw that the two equations,
10 x = y and
x = log (y), said the same thing.
If we replace x in the first equation
by the x of the second equation we get this identity:
10 log( y) = y
and if we replace y in the second equation
by the y of the first equation we get this identity:
x = log (10 x )
These identities are useful for showing how the logarithm and antilogarithm cancel each other.
- If you compare the graph
of y = log (x) to the
graph
of y = 10 x
then you see that one can be gotten from the other by interchanging
the x and y axes. This always happens with inverse functions.
How to use the base 10 logarithm function in the Algebra Coach
- Type log(x) into the textbox, where x is the argument.
The argument must be enclosed in brackets.
- Set the relevant options:
- Set the exact / floating point option.
In floating point mode the base 10 logarithm of any number is evaluated.
In exact mode the base 10 logarithm of an integer is not evaluated because
doing so would result in an approximate number.
- Turn on complex numbers if you want to be able to evaluate the
base 10 logarithm of a negative or complex number.
- Click the Simplify button.
Algorithm for the base 10 logarithm function
Click here
to see the algorithm that computers use to evaluate the base 10 logarithm function.
The natural logarithm function
Background: You might find it useful to read the previous section
on the base 10 logarithm function before reading this section.
The two sections closely parallel each other.
Recall that the base 10 logarithm function takes a number like 16
as input, calculates that it can be written as 10 1.2,
and then returns the exponent 1.2 as its output value. But why use base 10?
After all, probably
the only reason that the number 10 is important to humans is that
they have 10 fingers with which they first learned to count.
Maybe on some other planet populated by 8-fingered beings they use base 8!
In fact probably the most important number in
all of mathematics (click here to see why)
is the number 2.71828…, which we give the name e,
in honor of Leonard Euler, who first discovered it.
It will be important to be able to take any positive number, y, and express
it as e raised to some power, x.
We can write this relationship in equation form:
y = e x
For example 5 can be written as e 1.6
(the exponent is approximate).
How do we know that this is the correct power of e?
Because we get it from the graph shown below.
To make this graph we made a table of a few obvious values of
y = e x as shown below, left.
Then we plotted the values in the graph (they are the red dots) and drew a smooth
curve through them. Then we observed that the curve went through y = 5
and x = 1.6 (the black dot). This means that 5 = e 1.6.
If you compare this graph to that of
y = 10 x
you see that both have the same so-called exponential growth shape
but that this graph grows more slowly.
We next define a function called the natural logarithm that takes a number like 5
as input, calculates that it can be written as e 1.6,
and returns the exponent 1.6 as its output value. Here is the formal definition.
Definition: The natural logarithm is the function that takes
any positive number x as input and returns the
exponent to which the base e must be raised to obtain x.
It is denoted ln(x). (e denotes the number 2.71828…)
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Note that to avoid confusion the natural logarithm function is denoted
ln(x) and
the base 10 logarithm function is denoted log(x) .
Example 1: Evaluate ln ( e 4.7 ).
The argument of the
natural logarithm function is already expressed
as e raised to an exponent, so the natural logarithm function simply returns the exponent.
ln ( e 4.7 ) = 4.7
Example 2: Evaluate ln ( 5 ).
According to the graph 5 = e 1.6. The logarithm function returns the exponent 1.6.
ln ( 5 ) = 1.6
Example 3: Evaluate ln ( e ).
Express the argument
as e raised to the exponent 1 and return the exponent.
ln ( e ) = ln ( e 1 ) = 1
Example 4: Evaluate ln ( 1 ).
Express the argument
as e raised to the exponent 0 and return the exponent.
ln ( 1 ) = ln ( e 0 ) = 0
Graph: The red curve shown to the right is the graph of the natural logarithm function, y = ln (x).
Notice that for any positive x it is single valued and for any negative x it is undefined.
If you compare this graph to the
graph of y = e x
then you see that one can be gotten from the other by interchanging
the x and y axes.
For comparison the blue curve shows the
base 10 logarithm function, y = log (x). It has exactly the same shape but is only 43% as tall.
Domain and range:
The domain of the natural logarithm function is all positive real numbers and the range is
all real numbers.
The natural logarithm function can be
extended to the complex numbers,
in which case the domain is all complex numbers except zero.
The natural logarithm of zero is always undefined.
Solving the Equation e x = c for x
by using the natural logarithm function
Suppose that x is unknown but that e x equals
a known value c. Then finding x requires solving this equation
for x:
e x = c
The solution is
x = ln (c)
because finding ln (c) means expressing
c as e raised to an exponent, and then returning that exponent.
But the original equation says that that exponent is x.
Note that if c is negative then there is no real solution.
However there is a complex solution.
Furthermore if c = 0 then there is no solution at all.
ln (x) and e x are inverse functions
Consider the e x function which takes x and returns
e x, like this:
The natural logarithm function is defined to do exactly the opposite, namely:
Therefore these are inverse functions.
Note the following:
- We saw above that the solution of
e x = y is
x = ln (y). We should look at these two equations
as expressing the same relationship between x and y but
from different points of view. The first equation is the relationship solved for
y and the second one is the relationship solved for x.
(An analogy is that the statement “Tom is Jane’s brother”
is equivalent to the statement that “Jane is Tom’s sister”.)
- In the previous bullet we saw that the two equations,
e x = y and
x = ln (y), said the same thing.
If we replace x in the first equation
by the x of the second equation we get this identity:
e ln (y) = y
and if we replace y in the second equation
by the y of the first equation we get this identity:
x = ln (e x )
These identities are useful for showing how the natural logarithm and e x
functions cancel each other.
- If you compare the graph
of y = ln (x) to the
graph
of y = e x
then you see that one can be gotten from the other by interchanging
the x and y axes. This always happens with inverse functions.
How to use the natural logarithm function in the Algebra Coach
- Type ln(x) into the textbox, where x is the argument.
The argument must be enclosed in brackets.
- Set the relevant options:
- Set the exact / floating point option.
In floating point mode the natural logarithm of any number is evaluated.
In exact mode the natural logarithm of an integer is not evaluated because
to do so would result in an approximate number.
- Set the e does / does not represent 2.718… option.
(Set this to does represent
so that for example ln(e 3 ) simplifies to 3.)
- Turn on complex numbers if you want to be able to evaluate the
natural logarithm of a negative or complex number.
- Click the Simplify button.
Algorithm for the natural logarithm function
Click here
to see the algorithm that computers use to evaluate the natural logarithm function.
The exponential function (with base e, the e x function)
Background: You might find it useful to read the previous section
on the natural logarithm function before reading this section.
There we saw that it is possible to use the number e
(which is approximately 2.71828…) as a base and to raise it to
any power, x, and produce any positive number y.
We can write this relationship in equation form:
y = e x
Here is a graph of y = e x (the blue curve).
For comparison we also show graphs of y = 2 x
and y = 4 x.
Because the number e is between 2 and 4
the curve y = e x lies between the curves
y = 2 x and
y = 4 x.
All three of these curves are called exponential functions because the
independent variable x is in the exponent. All three have the property
that the higher up the curve you go the steeper they get. However
y = e x has the special property that at every
point along the curve the slope equals the height.
This property concerning slopes makes it a very important function in
calculus.
Note: In this section when we say ‘the exponential function’ we mean the one with base e.
Definition: We define a function called the exponential function (denoted exp) that takes an argument x
and returns the value of e raised to the power x:
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You can think of exp(x) as just an alternative (functional) notation for
the expression e x.
So of course the functional form exp(x) has all the
properties that the exponential form e x has.
The Algebra Coach has an option that allows you to use one form or the other.
Here is a table comparing the “look” of the various properties in the two forms:
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Graph: The blue curve is the graph of
y = e x
(i.e. of the exponential function).
It has the property that its slope equals its height everywhere.
The dotted red lines show the slope of the curve at various points along
the curve. Notice that the slope is 5 when the height is 5, and so on.
If you compare this graph of the exponential function to the
graph of the natural logarithm function
then you see that one can be gotten from the other by interchanging
the x and y axes.
Domain and range:
The domain of the exponential function is all real numbers and the range is
all positive real numbers.
The exponential function can be
extended to the complex numbers,
in which case the domain and the range is all complex numbers.
Solving the Equation ln (x) = c for c
by using the exponential function
Suppose that x is unknown but that ln (x) equals
a known value c. Then finding x requires solving this equation
for x:
ln (x) = c.
The solution is:
x = e c,
or
x = exp (c).
This was explained in the previous section on the natural logarithm function.
ln (x) and e x are inverse functions
This was explained in the previous section on the natural logarithm function.
How to use the exponential function in the Algebra Coach
- Type exp(x) into the textbox, where x is the argument.
The argument must be enclosed in brackets.
- Set the relevant options:
- Set the exponential function option to this:
- Set the exact / floating point option.
In floating point mode exp(r) for any number r is evaluated.
In exact mode exp(i) for any integer i is not evaluated because
to do so would result in an approximate number.
- Turn on complex numbers if you want to be able to evaluate
exp(c) for a complex number c.
- Click the Simplify button.
Note: You never need to use the functional
notation exp (x) in the Algebra Coach.
You can always use the notation e x instead.
In fact the default setting for the exponential function option is
exp (x) → e x.
Algorithm for the e x or exponential function
Click here
to see the algorithm that computers use to evaluate
e x or exp (x).
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