base 10 logarithm (log) | natural logarithm (ln) | exponential function (exp or e^{x }) |
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). |
log ( 1000 ) = log ( 10^{ 3} ) = 3
log ( 10^{ 5.7} ) = 5.7
log ( 16 ) = log ( 10^{ 1.2} ) = 1.2
10^{ x} = cThe 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.
10^{ log( y)} = yand 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.
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.
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…) |
ln ( e^{ 4.7} ) = 4.7
ln ( 5 ) = 1.6
ln ( e ) = ln ( e^{ 1} ) = 1
ln ( 1 ) = ln ( e^{ 0} ) = 0
e^{ x} = cThe 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.
e^{ ln (y)} = yand 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.
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}.
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: |
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.