the log function     the ln function     the exp function  


The log function

Background: Any positive number, y, can be written as 10 raised to some power, x. We can write this relationship in equation form:
y = 10 x
For example it is obvious that 1000 can be written as 10 3. It is not so obvious that 16 can be written as 10 1.2. How do we know that this is the correct power of 10? Because we get it from the graph shown below.

To make this graph we made a table of a few obvious values of  y = 10 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 = 16 and x = 1.2 (the black dot). This means that 16 = 10 1.2.


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We next define a function called the logarithm that takes a number like 16 as input, calculates that it can be written as 10 1.2, and returns the exponent 1.2 as its output value. Here is the formal definition.


Definition: log(x) is defined as 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.



Example 1:   Evaluate log ( 10 5.7 ). In this example the argument of the log function is already expressed as 10 raised to an exponent, so the log function simply returns the exponent:
log ( 10 5.7 ) = 5.7

Example 2:   Evaluate log ( 1000 ). The argument is a number that is easily expressed as 10 raised to an exponent. We do this and the log function then returns the exponent:
log ( 1000 ) = log ( 10 3 ) = 3

Example 3:   Evaluate log ( 16 ). The argument is a number which we don't know how to express as 10 raised to an exponent (unless we remember the above discussion which said that 16 = 10 1.2 ). Therefore we use a calculator or the Algebra Coach to evaluate it:
log ( 16 ) = 1.2

Example 4:   log ( x ). The argument is an expression. Until we can evaluate that expression we must leave this logarithm as is.




Graph: The blue curve shown to the right is the graph of the log function. Notice that for any positive x it returns a single value. For any negative x it is undefined. If you compare this graph of the log function 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.

For comparison we have also shown the graph of the ln function (the natural log function) in red. The ln graph has the same shape as the log graph but is 2.3 times as tall. The ln function is described below.

An important feature of the log function is that it increases very slowly as x becomes very large. It describes nicely how the human ear percieves loud sounds and the way the human eye percieves bright lights.



Domain and range: The domain of the log function is all positive real numbers and the range is all real numbers.

The log function can be extended to the complex numbers, in which case the domain is all complex numbers except zero. The logarithm of zero is always undefined.




Some special values of the log function





Solving the Equation 10 x = y for x by using the log function

Suppose that x is unknown but that 10 x equals a known value y. Then finding x requires solving the following equation for x:
10 x = y
The solution is:
x = log (y)
This is because finding log (y) means expressing y as 10 to an exponent, and then returning that exponent. But the original equation says that that exponent is x. Note that if y is negative then there is no real solution. However there is a complex solution. Furthermore if y = 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 log function is defined to do exactly the opposite, namely:
Therefore these are inverse functions.

Note the following:


How to use the log function in the Algebra Coach


Algorithm for the log function

Click here to see the algorithm that computers use to evaluate the log function.




The ln function

Background: You might find it useful to read the previous section on the log function before reading this section. Recall that the log 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 they use base 8!

In fact probably the most important number in all of mathematics (click here to see why) is the number 2.7182818284590…, 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.


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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: ln(x) is defined as 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. (e denotes the number 2.7182818284590…)


From now on, to avoid confusion, we will call ln(x) the natural logarithm function and log(x) the base 10 logarithm function.


Example 1:   Evaluate ln ( e 4.7 ). In this example the argument of the ln function is already expressed as e raised to an exponent, so the ln function simply returns the exponent:
ln ( e 4.7 ) = 4.7

Example 2:   Evaluate ln ( 8.6 ). The argument is a number which we don't know how to express as e raised to an exponent . Therefore we use a calculator or the Algebra Coach to evaluate it:
ln ( 8.6 ) = 2.15

Example 3:   ln ( x ). The argument is an expression. Until we can evaluate that expression we must leave this natural logarithm as is.


Example 4:   Evaluate ln ( e ). Express the argument as e raised to the exponent 1 and return the exponent:
ln ( e ) = ln ( e 1 ) = 1

Example 5:   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 ln function. Notice that for any positive x it returns a single value. For any negative x it is undefined. If you compare this graph of the ln function 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 we have also shown the graph of the base 10 log function in blue. The log graph has the same shape as the ln graph but is only 43% as tall.



Domain and range: The domain of the ln function is all positive real numbers and the range is all real numbers.

The ln 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 = y for x by using the ln function

Suppose that x is unknown but that e x equals a known value y. Then finding x requires solving this equation for x:
e x = y
The solution is:
x = ln (y)
because finding ln (y) means expressing y as e to an exponent, and then returning that exponent. But the original equation says that that exponent is x. Note that if y is negative then there is no real solution. However there is a complex solution. Furthermore if y = 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 ln function is defined to do exactly the opposite, namely:
Therefore these are inverse functions.

Note the following:


How to use the ln function in the Algebra Coach


Algorithm for the ln function

Click here to see the algorithm that computers use to evaluate the natural logarithm function.




The exp or e x function

Background: You might find it useful to read the previous section on the ln function before reading this section. There we saw that it is possible to use the number e (which is approximately 2.7182818284590…) 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.



Definition: We define a function called the exp function that takes an argument x and returns the value of e raised to the power x:

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:
Functional form Exponential form
info on this property
info on this property
info on this property
info on this property
info on this property
info on this property
info on this property
info on this property




Graph: The blue curve is the graph of  y = e x  (i.e. of the exp 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 exp function to the graph of the ln 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 exp function is all real numbers and the range is all positive real numbers.

The exp 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) = y for x by using the e x or exp function

Suppose that x is unknown but that ln (x) equals a known value y. Then finding x requires solving this equation for x:
ln (x) = y.
The solution is:
x = e y,    or    x = exp (y).
This was explained in the previous section on the ln function.




ln (x) and e x are inverse functions

This was explained in the previous section on the ln function.



How to use the exp function in the Algebra Coach 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 exp function

Click here to see the algorithm that computers use to evaluate e x or exp (x).