Natural Logarithm
Thenatural logarithmof a numberis thelogarithmto the base , whereis themathematical constantapproximately equal to. It is usually written using the shorthand notation , instead ofas you might expect.You can rewrite a natural logarithm inexponential formas follows:
Example 1:
Find.
在一个科学entific calculator, you can simply pressfollowed byto get the answer: approximately.
The exponential form of the equation you're solving is
Example 2:
Solve the equation. Round to the nearest thousandth.
First, rewrite the equation in exponential form.
Use a calculator. (Most scientific calculators have a button which gives a good approximation for; if yours doesn't have one, use.)
The usualproperties of logarithmsare also true for the natural logarithm.
Example 3:
Simplify.
The following property lets you simplify logarithms of a power:
So,
Now use the property that the log of a product is equal to the sum of the logs.
So,
Thegraph of the logarithmic function (shown in blue, below) looks similar to the graphs of related functionsor(remember that if no base is written, the base of the logairthm is understood to be).
The function has an asymptote atand an-intercept at. It passes through the pointsand.