the log function | the ln function | the exp function |
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.
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. |
log ( 10^{ 5.7} ) = 5.7
log ( 1000 ) = log ( 10^{ 3} ) = 3
log ( 16 ) = 1.2
10^{ x} = yThe 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.
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 log and antilog functions 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: 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…) |
ln ( e^{ 4.7} ) = 4.7
ln ( 8.6 ) = 2.15
ln ( e ) = ln ( e^{ 1} ) = 1
ln ( 1 ) = ln ( e^{ 0} ) = 0
e^{ x} = yThe 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.
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 ln 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) = y.The solution is:
x = e^{ y}, or x = exp (y).This was explained in the previous section on the ln function.