Victorian Curriculum Year 10A - 2020 Edition

5.04 Logarithms

Lesson

We've seen equations like $y=B^x$`y`=`B``x` 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 $B^x$`B``x`, 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$3`x`=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 $\log_By$`l``o``g``B``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`=`B``x`

then

$\log_By=x$`l``o``g``B``y`=`x`

In other words, $\log_By$`l``o``g``B``y` is the number of $B$`B` factors that multiply together to make $y$`y`. It follows that $\log_381=4$`l``o``g`381=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 $\log y=\log_10y$`l``o``g``y`=`l``o``g`10`y`. If we wanted to find $\log81$`l``o``g`81, 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.

Summary

Logarithms are expressions of the form $\log_By$`l``o``g``B``y`, where $B$`B` is any number and $y$`y` is a pronumeral.

In $\log_By$`l``o``g``B``y`, $B$`B` is the base of the logarithm.

By convention, if the base is not specified then $B=10$`B`=10.

If $y=B^x$`y`=`B``x` then $\log_By=x$`l``o``g``B``y`=`x`, so $y$`y` is the number of $B$`B` factors that are multiplied together to give $x$`x`.

Rewrite the equation $9^x=81$9`x`=81 in logarithmic form (with the index as the subject of the equation).

Evaluate $\log_8\left(\frac{1}{64}\right)$`l``o``g`8(164).

Evaluate $\log_{10}$`l``o``g`10$45$45.

Round your answer to two decimal places.

Use the definition of a logarithm to establish and apply the laws of logarithms and investigate logarithmic scales in measurement