6.01 Logarithms

We are already familiar with exponential functions. In this chapter, our focus will be on a closely related function called the logarithmic function. The logarithmic function provides an alternate way to write exponential expressions and is in fact the inverse function of the exponential function (we will explore the idea of inverse functions further in our next lesson).

Note: We read logarithms such as $\log_28$log2​8 as "the log base $2$2 of $8$8".

A logarithm of the form $\log_ba$logb​a is equal to the index to which we would raise the base $b$b in order to obtain the argument $a$a.

The table below shows some examples of converting between exponential form and logarithmic form:

Exponential	Logarithm
$2^3=8$23=8	$\log_28=3$log2​8=3
$5^{-2}=\frac{1}{25}$5−2=125​	$\log_5\frac{1}{25}=-2$log5​125​=−2
$10^0=1$100=1	$\log_{10}1=0$log10​1=0

 

 

Worked examples

Example 1

Rewrite $6^2=36$62=36 in logarithmic form.

Think: In this exponential expression we can identify the base as $6$6, the index as $2$2 and the result as $36$36. To transpose it into a logarithmic expression, $\log_ba=x$logb​a=x, $b$b is the base, $a$a the argument is the result of the exponential equation, and $x$x is the index from the exponential equation.

Do: Thus we have $\log_636=2$log6​36=2. Which we can read as "the index that $6$6 must be raised to in order to obtain $36$36 is $2$2"

Example 2

Rewrite $\log_{\frac{1}{4}}\frac{1}{64}=3$log14​​164​=3 in exponential form.

Think: From this logarithmic expression we can identify the components of the exponential expression: the base $\frac{1}{4}$14​, the index $3$3 and the result $\frac{1}{64}$164​.

Do: Thus we have $\left(\frac{1}{4}\right)^3=\frac{1}{64}$(14​)3=164​

 

Practice questions

question 1

question 2

 

Evaluating logarithmic expressions

Recall that the log expression $\log_ba$logb​a is the number that the base $b$b must be raised to in order to result in $a$a. This is a key understanding of the expression.

For example, to evaluate $\log_{10}1000$log10​1000, we ask ourselves "to what index must $10$10 be raised to get a result of $1000$1000?" Since $10$10 raised to the index $3$3 would result $1000$1000, then $3$3 is said to be the base $10$10 logarithm of $1000$1000. That is $\log_{10}1000=3$log10​1000=3

Consider the square root function, where we can think of the expression $\sqrt{a}$√a both as an operation and a number in its own right. For example $\sqrt{4}$√4 is the positive number which, when multiplied by itself, will obtain $4$4. We can simplify the expression by evaluating the square root, since $\sqrt{4}=2$√4=2. However, some expressions cannot be simplified $\sqrt{5}$√5 is a number when multiplied by itself obtains $5$5 but while we could obtain an approximation using our calculator we cannot express the number exactly without use of the $\sqrt{\ }$√  symbol.

Similarly a logarithmic expression can be considered both an operation or number in its own right. For example $\log_28$log2​8 is the index $2$2 must be raised to in order to get a result of $8$8. We can simplify the expression by evaluating the logarithm, since $2^3=8$23=8, $\log_28=3$log2​8=3.

Some expressions cannot be simplified though - for instance, $\log_310$log3​10 is the index $3$3 must be raised in order to get a result of $10$10. What would that be? Since $3^2=9$32=9 and $3^3=27$33=27 it must be a little larger than $2$2 and we could obtain an approximation using our calculator $\log_310\approx2.0959$log3​10≈2.0959. If we want to express the number exactly, we would simply leave it as $\log_310$log3​10.

 

Worked example

question 3

Evaluate without the use of technology $\log_5125$log5​125.

Think: What index must $5$5 be raised to get a result of $125$125? It can be helpful to rewrite $125$125 as a power of $5$5 first, then write down the index.

Do:

$\log_5125$log5​125	$=$=	$\log_55^3$log5​53	Rewrite $125$125 as a power of $5$5.
	$=$=	$3$3	Simplify by just writing the index.

 

Alternatively, we could rewrite the question using a pronumeral as the result, then transpose into an exponential expression and solve for $x$x.

Do: $\log_5125=x$log5​125=x

$5^x$5x	$=$=	$125$125	Transposing into an exponential expression.
$5^x$5x	$=$=	$5^3$53	Making the bases the same on each side.
$x$x	$=$=	$3$3	Equating the indices.

Hence $\log_5125=3$log5​125=3.

Choosing the model of your calculator below to see how to evaluate a logarithmic expression using technology.

Casio ClassPad

TI Nspire

 

Practice questions

question 3

Question 4

Question 5