3.06 Logarithm laws and change of base
Because of the link between indices and logarithms, it follows that these index laws can be manipulated to create a set of equivalent laws that apply to logarithms.
The addition and subtraction logarithm laws
The addition law of logarithms relates the sum of two logarithms to the logarithm of a product.
Similarly, the subtraction law of logarithms relates the difference of two logarithms to the logarithm of a quotient.
The addition law of logarithms: When adding logs with the same base, multiply the numbers. That is,
$\log_bx+\log_by=\log_b\left(xy\right)$logbx+logby=logb(xy)
The subtraction law of logarithms: When subtracting logs with the same base, divide the numbers. That is,
$\log_bx-\log_by=\log_b\left(\frac{x}{y}\right)$logbx−logby=logb(xy)
We can prove these laws by using the corresponding properties of exponentials:
Proof of addition law:
We start by letting $\log_bx=N$logbx=N and $\log_by=M$logby=M. We can rewrite these two equations in their equivalent exponential forms, $b^N=x$bN=x and $b^M=y$bM=y. Multiplying these two expressions gives us the result:
| $xy$xy | $=$= | $b^N\times b^M$bN×bM |
Writing down the product |
| $=$= | $b^{N+M}$bN+M |
Using a property of exponentials |
|
| $\log_b\left(xy\right)$logb(xy) | $=$= | $N+M$N+M |
Rewriting in logarithmic form |
| $\log_b\left(xy\right)$logb(xy) | $=$= | $\log_bx+\log_by$logbx+logby |
Substituting |
Proof of subtraction law:
We follow a similar procedure and start by letting $\log_bx=N$logbx=N and $\log_by=M$logby=M. We can rewrite these in their exponential forms, $b^N=x$bN=x and $b^M=y$bM=y. Taking the quotient of the two expressions gives us the result:
| $\frac{x}{y}$xy | $=$= | $\frac{b^N}{b^M}$bNbM |
Writing down the quotient |
| $=$= | $b^{N-M}$bN−M |
Using a property of exponentials |
|
| $\log_b\left(\frac{x}{y}\right)$logb(xy) | $=$= | $N-M$N−M |
Rewriting in logarithmic form |
| $\log_b\left(\frac{x}{y}\right)$logb(xy) | $=$= | $\log_bx-\log_by$logbx−logby |
Substituting |
These two laws are especially valuable if we want to simplify expressions or solve equations involving logarithms.
Worked examples
example 1
Simplify the logarithmic expression $\log_310-\log_32$log310−log32.
Think: Since the two logarithms have the same base, we can use the subtraction law of logarithms.
Do: To use the subtraction property of two logarithms, we can divide the arguments:
| $\log_310-\log_32$log310−log32 | $=$= | $\log_3\left(\frac{10}{2}\right)$log3(102) |
Using the subtraction law |
| $=$= | $\log_35$log35 |
Simplifying the argument |
example 2
Rewrite $\log_56x$log56x as a sum or difference of two logarithms.
Think: Since there is a product in the logarithm, we can use the addition law in reverse.
Do: So using the addition law, we can rewrite $\log_56x$log56x in the form:
$\log_56+\log_5x$log56+log5x
Reflect: Note that we could have chosen any pair of factors that multiply to give $6x$6x to rewrite this expression. So, for example, another possible answer would be
$\log_52+\log_53x$log52+log53x
Practice questions
question 1
Simplify each of the following expressions without using a calculator. Leave answers in exact form.
$\log_{10}11+\log_{10}2+\log_{10}9$log1011+log102+log109
$\log_{10}12-\left(\log_{10}2+\log_{10}3\right)$log1012−(log102+log103)
question 2
Express $\log\left(\frac{pq}{r}\right)$log(pqr) as the sum and difference of log terms.
The power logarithm law
From previous lessons we know that $x=a^m$x=am and $m=\log_ax$m=logax are equivalent. We are able to use this definition to discover some more helpful properties of logarithms such as the power law.
Exploration
Let's simplify $\log_a\left(x^2\right)$loga(x2) using the logarithmic properties that we already know.
| $\log_a\left(x^2\right)$loga(x2) | $=$= | $\log_a\left(x\times x\right)$loga(x×x) |
Rewrite $x^2$x2 as a product, $x\times x$x×x. |
| $=$= | $\log_ax+\log_ax$logax+logax |
Use the addition law of logarithms, |
|
| $=$= | $2\log_ax$2logax |
Collect logarithms with the same base and variables. |
We can also simplify logarithms with powers using the power law of logarithms, this property can be used for any values of the power $n$n.
The power law of logarithm: When the number in the log is raised to a power, we can bring that power down to the front to be multiplied to the log.
$\log_a\left(x^n\right)=n\log_ax$loga(xn)=nlogax
Now, let's simplify $\log_a\left(x^2\right)$loga(x2) using the power law.
| $\log_a\left(x^2\right)$loga(x2) | $=$= | $2\log_ax$2logax |
Notice this gives the exact same result as using the addition law.
Let's consider the proof of the power law of logarithms:
| Proof | |||||
| Let $x$x | $=$= | $a^m$am | |||
| $x^n$xn | $=$= | $\left(a^m\right)^n$(am)n |
Raise both sides of $x=a^m$x=am to the power $n$n. |
||
| $x^n$xn | $=$= | $a^{mn}$amn |
Use the index law $\left(a^m\right)^n=a^{mn}$(am)n=amn. |
||
| $\log_a\left(x^n\right)$loga(xn) | $=$= | $mn$mn |
Express as a logarithm. |
||
| $\log_a\left(x^n\right)$loga(xn) | $=$= | $n\log_ax$nlogax |
Substitute back for $m=\log_ax$m=logax. |
||
Practice questions
question 3
Use the properties of logarithms to rewrite the expression $\log_4\left(x^7\right)$log4(x7).
Write your answer without any powers.
Question 4
Use the properties of logarithms to rewrite the expression $\log\left(\left(x+6\right)^5\right)$log((x+6)5).
Write your answer without any powers.
Question 5
Evaluate $\log_5125^{\frac{5}{4}}$log512554.
Using a combination of laws to simplify logarithmic expressions
So far we've seen some laws of logarithms in isolation, and looked at how they may be individually useful to simplify an expression or solve an equation.
However, sometimes simplifying an expression may require using several of these laws.
The laws we have looked at are summarised below.
The addition law of logarithms is given by:
$\log_bx+\log_by=\log_b\left(xy\right)$logbx+logby=logb(xy)
The subtraction law of logarithms is given by:
$\log_bx-\log_by=\log_b\left(\frac{x}{y}\right)$logbx−logby=logb(xy)
The power law of logarithms is given by:
$\log_b\left(x^n\right)=n\log_bx$logb(xn)=nlogbx
Some useful identities of logarithms are:
$\log_b1=0$logb1=0 and $\log_bb=1$logbb=1
Worked example
Example 3
Simplify the expression $\log_3\left(100x^3\right)-\log_3\left(4x\right)$log3(100x3)−log3(4x), writing your answer as a single logarithm.
Think: Each logarithm in the expression has the same base, so we can express the difference as a single logarithm using the subtraction law.
Do: To use the subtraction law, we take the quotient of the two arguments as follows:
| $\log_3\left(100x^3\right)-\log_3\left(4x\right)$log3(100x3)−log3(4x) | $=$= | $\log_3\left(\frac{100x^3}{4x}\right)$log3(100x34x) |
Using the subtraction law |
| $=$= | $\log_3\left(25x^2\right)$log3(25x2) |
Simplifying the argument |
|
| $=$= | $\log_3\left(\left(5x\right)^2\right)$log3((5x)2) |
Rewriting the argument as a power |
|
| $=$= | $2\log_3\left(5x\right)$2log3(5x) |
Using the power law |
Practice questions
question 6
Simplify each of the following expressions without using a calculator. Leave answers in exact form.
$\log_{10}10+\frac{\log_{10}\left(15^{20}\right)}{\log_{10}\left(15^5\right)}$log1010+log10(1520)log10(155)
$\frac{8\log_{10}\left(\sqrt{10}\right)}{\log_{10}\left(100\right)}$8log10(√10)log10(100)
question 7
Express $5\log x+3\log y$5logx+3logy as a single logarithm.
Question 8
Using the rounded values $\log_{10}7=0.845$log107=0.845 and $\log_{10}3=0.477$log103=0.477, calculate $\log_{10}49+\log_{10}27$log1049+log1027 to three decimal places.
Changing the base
We often encounter occasions where we need to take a logarithm given in one base and express it as a logarithm in another base. For example, if we wanted to use the calculator to evaluate a logarithm but need to change the base to $10$10 or $e$e. A change of base formula has been developed to do just that.
Suppose we think of a number $y$y expressed as $y=\log_pa$y=logpa.
Since $y=\log_pa$y=logpa, then from the definition of a logarithm this means that:
| $a$a | $=$= | $p^y$py |
If we now take logarithms of some other base $q$q, we have that:
| $\log_qa$logqa | $=$= | $\log_q\left(p^y\right)$logq(py) |
| $\log_qa$logqa | $=$= | $y\log_qp$ylogqp |
From the working rules of logarithms this simplifies to:
| $y$y | $=$= | $\frac{\log_qa}{\log_qp}$logqalogqp |
Look carefully at this result. It is saying that:
| $\log_pa$logpa | $=$= | $\frac{\log_qa}{\log_qp}$logqalogqp |
For example:
| $\log_58$log58 | $=$= | $\frac{\log_{10}8}{\log_{10}5}$log108log105 |
So if we knew the common logs of $8$8 and $5$5, we could determine $\log_58$log58. Now, to five decimal places:
| $\log_{10}8$log108 | $=$= | $0.90309$0.90309 |
| $\log_{10}5$log105 | $=$= | $0.69897$0.69897 |
Then we have
| $\log_58$log58 | $\approx$≈ | $\frac{0.90309}{0.69897}$0.903090.69897 |
| $=$= | $1.29203$1.29203 ($5$5 d.p.) |
Note that $\log_58$log58 could also be expressed as $\frac{\log_2\left(8\right)}{\log_2\left(5\right)}$log2(8)log2(5) which would give the same answer $1.29203$1.29203. Any base can be used, now we have the above relationship.
As another example:
| $\log_b10$logb10 | $=$= | $\frac{\log_{10}10}{\log_{10}b}$log1010log10b |
| $\log_b10$logb10 | $=$= | $\frac{1}{\log_{10}b}$1log10b |
And:
| $\log_b10\times\log_{10}b$logb10×log10b | $=$= | $1$1 |
This is an interesting result. We can generalise this to show that:
| $\log_ba\times\log_ab$logba×logab | $=$= | $1$1 |
So that $\log_ba$logba and $\log_ab$logab are multiplicative inverses.
To change the base of a logarithm, we do $\frac{\text{log of the number}}{\text{log of the base}}$log of the numberlog of the base, that is:
$\log_ab=\frac{\log_cb}{\log_ca}$logab=logcblogca
Practice questions
Question 9
Rewrite $\log_416$log416 in terms of base $10$10 logarithms.
Question 10
Rewrite $\log_3\sqrt{5}$log3√5 in terms of base $10$10 logarithms.