6.01 Definition of the logarithm
When we studied index laws there were other names for the index of a number, such as power and exponent. In this topic we will refer to expressions with an index as exponentials.
In mathematics we have pairs of inverse functions, or opposite operations. For example, addition and subtraction are inverses, as are multiplication and division. The inverse function of an exponential is called a logarithm. A logarithm "undoes" an exponential and the reverse is true.
If we have an exponential equation of the form
$b^x=a$bx=a,
then we can rewrite it a a logarithmic equation of the form
$\log_ba=x$logba=x.
In this equation, $b$b is called the base, just like in exponentials; $a$a is called the argument, and $x$x is the answer (or the power)
Note: we read logarithms like $\log_28$log28 as "the log base $2$2 of $8$8".
We use logarithms when we are interested in finding out the index ($x$x) to which we would raise the base ($b$b) in order to obtain the argument ($a$a).
When evaluating logarithms in the form $\log_ba$logba, we ask "to what power must base $b$b be raised to get a result of $a$a?"
The table below shows some examples of converting between exponential form and logarithmic form:
| Exponential | Logarithm |
|---|---|
| $2^3=8$23=8 | $\log_28=3$log28=3 |
| $5^{-2}=\frac{1}{25}$5−2=125 | $\log_5\frac{1}{25}=-2$log5125=−2 |
| $10^0=1$100=1 | $\log_{10}1=0$log101=0 |
The result of any exponential $b^x$bx is a positive number. Correspondingly, this means that we can only take the logarithm of a positive argument. We also only talk about logarithms with a positive base (think about why this might be the case). Additionally, since $1^x=1$1x=1 for any $x$x, we cannot take a logarithm with a base of $1$1 (an inverse doesn't make sense here).
In summary:
$\log_ba$logba only makes sense when $a>0$a>0 and $b>0$b>0, $b\ne1$b≠1
For example, we cannot evaluate $\log_2(-4)$log2(−4)nor $\log_{-3}27$log−327
Worked examples
Example 1
Rewrite $6^2=36$62=36 in logarithmic form.
Think: Remember we want the index on its own and the logarithm on the other side
Do: $6$6 is the base and $2$2 is the index, so:
$\log_636=2$log636=2
Example 2
Rewrite $\log_432=2.5$log432=2.5 in exponential form.
Think: Remember that in exponential form we want isolate the resulting number after having the base raised to the index
Do: $32$32 is the result after raising the base $4$4 to the index $2.5$2.5, so:
$4^{2.5}=32$42.5=32
Practice questions
question 1
Evaluate $\log_216$log216.
Question 2
Rewrite the equation $3^x=81$3x=81 in logarithmic form (with the index as the subject of the equation).
Question 3
Rewrite the equation $\log_8x=6$log8x=6 in exponential form. (Do not solve for $x$x.)
Logarithms on the calculator
Scientific calculators almost always have a $\log$log button. This button allows us to calculate logarithms with base $10$10. Base $10$10 logarithms are also known as common logarithms, and these logarithms are sometimes written as simply $\log$log without writing the base of $10$10.
There will also be a "$\ln$ln" button, which is for natural logarithms (logarithms with base $e$e - more on this number later). As a second function to those two buttons, there will be the inverse (opposite) functions. $\log_{10}x$log10x and $10^x$10x are the inverse of each other and $\log_ex$logex and $e^x$ex are the inverse of each other.
The logarithm buttons on a scientific calculator
Worked example
Example 3
Write $10^x=650$10x=650 in logarithmic form and use the log button on the calculator to approximate $x$x to four significant figures.
Do: In logarithmic form the equation becomes $x=\log_{10}650$x=log10650
Using the calculator, we type "$\log$log" and "$650$650" then "$=$=" to evaluate $x$x. This gives $2.81291\ldots$2.81291…
To four significant figures, we have $x=2.812$x=2.812.
Using the definition of a logarithm
We have seen that $\log_ba=x$logba=x and $b^x=a$bx=a are inverse functions. That means that when the two functions are applied to a number one after the other, it results in the original number. For example, we know that $\log$log and $10^x$10x are inverse functions and we can check using the calculator that $\log_{10}10^4=4$log10104=4 and $10^{\log_{10}4}=4$10log104=4
This is true for all bases because of the definition of a logarithm. For example:
$\log_22^8=8$log228=8 because $8$8 is the power when $2^8$28 is written in index form.
$2^{\log_216}=16$2log216=16 because $\log_216$log216 (which equals $4$4) is the power when $16$16 is written in index form with base $2$2.
Therefore it leads to these identities:
$\log_aa^x=x$logaax=x and $a^{\log_ax}=x$alogax=x
for all $x>0$x>0 and $a>1$a>1
Worked example
Example 4
Simplify:
(a) $\log_99^{12}$log9912
Using the relationship between exponentials and logarithms, we know that $\log_99^{12}=12$log9912=12.
(b) $6^{\log_65}$6log65
Using the relationship between exponentials and logarithms, we know that $6^{\log_65}=5$6log65=5.