4.06 Logarithms

The logarithmic function provides an alternate way to write exponential expressions and is in fact the inverse function of the exponential function.

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.

Check below to see how to evaluate a logarithmic expression using technology.

TI Nspire

 

Practice questions

question 3

Question 4

Question 5