5.04 Logarithms
Introduction
We've seen equations like y=B^x before. It's straightforward enough to find y when we know x, but is it possible to find x if we know y?
The expression B^x, if x is a natural number, means the number of B factors multiplied together is x. So to find x in 3^x=81 we ask how many 3 factors are in 81, and the answer is 4. But we saw from exponential graphs that x can in general be any real number, including irrational numbers. In that case it doesn't make sense to multiply B \, x times.
Ideas
Logarithms
Logarithms are expressions of the form \log_{B}y, where B is some number and y is a pronumeral. B is called the base of the logarithm. The definition of a logarithm is that if y=B^x then \log_{B}y=x.
In other words, \log_{B}y is the number of B factors that multiply together to make y. It follows that \log_{3}81=4, since 3^4=81.
Of course, the value of the logarithm could be any real number. We will soon see how to find the exact values of logarithms, but we can approximate the value using a calculator.
First note that by convention, if B is not specified that means it has a base of 10. So \log_y=\log_{10}y.
If we wanted to find \log_{}81, then we can press the "log" button on a calculator and then enter 81. This gives us 1.908 to three decimal places.
Examples
Example 1
Rewrite the equation 9^x=81 in logarithmic form (with the index as the subject of the equation).
Example 2
Evaluate \log_{8} \left(\dfrac{1}{64}\right).
Example 3
Evaluate \log_{10}45. Round your answer to two decimal places.
Logarithms are expressions of the form \log_{B}y, where B is any number and y is a pronumeral.
In \log_{B}y, \, B, is the base of the logarithm.
By convention, if the base is not specified then B=10.
If y=B^x then \log_{B}y=x, so y is the number of B factors that are multiplied together to give x.