11.06 Logarithms

We've seen equations like $y=B^x$y=Bx before. It's straightforward enough to find $y$y when we know $x$x, but is it possible to find $x$x if we know $y$y?

The expression $B^x$Bx, if $x$x is a natural number, means the number of $B$B factors multiplied together is $x$x. So to find $x$x in $3^x=81$3x=81 we ask how many $3$3 factors are in $81$81, and the answer is $4$4. But we saw from exponential graphs that $x$x can in general be any real number, including irrational numbers. In that case it doesn't make sense to multiply $B$B $x$x times.

Logarithms are expressions of the form $\log_By$logB​y, where $B$B is some number and $y$y is a pronumeral. $B$B is called the base of the logarithm. The definition of a logarithm is that if

$y=B^x$y=Bx

then

$\log_By=x$logB​y=x

In other words, $\log_By$logB​y is the number of $B$B factors that multiply together to make $y$y. It follows that $\log_381=4$log3​81=4.

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 $B$B is not specified that means a base of $10$10. So $\log y=\log_{10}y$logy=log10​y. If we wanted to find $\log81$log81, then we can press the "log" button on a calculator and then enter $81$81. This gives us $1.908$1.908 to three decimal places.