Lesson

We might be curious about the relationship between population and time, or the number of configurations of a number plate and its length. These relationships are either exponential or logarithmic, depending on which quantity we choose as the subject of the relationship.

From a description of these relationships, we want to be able to identify its logarithmic equation and use the equation to draw a graph, or to infer quantities in the relationship.

- Consider a number plate that has $n$
`n`characters where each character can be any one of the $26$26 letters or the $10$10 digits. How can we construct an equation that expresses $n$`n`in terms of the number of possible combinations, $x$`x`?

Well, in each position there are $36$36 possible characters available. Consider a number plate with $7$7 characters, like the one shown below. There are $36\times36\times36\times36\times36\times36\times36$36×36×36×36×36×36×36 possible combinations, which we can write more simply as $36^7$367 combinations.$7$7 *digit number plate*

In general, we can say that there are $36^n$36`n`total possible combinations for a number plate of length $n$`n`, and so we have the equation$x=36^n$

`x`=36`n`We can make $n$

`n`the subject by rewriting the equation in logarithmic form:$n=\log_{36}x$

`n`=`l``o``g`36`x`

This equation may be valuable when forecasting where we might need to increase the number of characters for a growing population.

- How many characters would be needed for a country with a population of $2$2 million?

A quick substitution will tell us the number of characters needed for $2$2 million different possible number plates:$n$ `n`$=$= $\log_{36}x$ `l``o``g`36`x`(Writing down the equation) $n$ `n`$=$= $\log_{36}2000000$ `l``o``g`362000000(Substituting) $n$ `n`$\approx$≈ $4.0487...$4.0487... (Evaluating) $n$ `n`$=$= $5$5 (Rounding up) **up**. This will account for more than $2$2 million possible number plates - if we had rounded down instead, there wouldn't be enough unique number plates for everyone in the population.

Now consider the graph of the logarithm function given below.Graph of $n=\log_{36}x$ `n`=`l``o``g`36`x` - If the population doubles every year, how many years will it take until they have to increase the number of characters used?

Well after one year, the population will be $4$4 million, then it will become $8$8 million and so on. One strategy is to first use a point on the graph to identify when $n$`n`first exceeds $5$5. Well we can see that graph roughly crosses $n=5$`n`=5 when $x=60$`x`=60 million.Point indicated at $x=60$ `x`=60 million

If we substitute $x=60$`x`=60 million into the equation to check, we get a value of $n=4.9978\dots$`n`=4.9978… which is very close to our value of $n=5$`n`=5, but falls a little short.

Since the population increases in powers of $2$2, we know that after $6$6 years, the population will be $2^6=64$26=64 million. Substituting $x=64$`x`=64 million into the equation we get$n$ `n`$=$= $\log_{36}64000000$ `l``o``g`3664000000(Substituting) $n$ `n`$\approx$≈ $5.015\dots$5.015… (Simplifying) $n$ `n`$=$= $6$6 (Rounding up)

So after $6$6 years, the population will increase to $x=64$`x`=64 million. At this point, licence plates that are only $5$5 digits long will not be enough to account for everyone in the population, and so the country will have to start issuing $6$6 digit licence plates.

pH is a measure of how acidic or alkaline a substance is, and the pH scale goes from $0$0 to $14$14, $0$0 being most acidic and $14$14 being most alkaline. Water in a stream has a neutral pH of about $7$7. The pH $\left(p\right)$(`p`) of a substance can be found according to the formula $p=-\log_{10}h$`p`=−`l``o``g`10`h`, where $h$`h` is the substance’s hydrogen ion concentration.

Store-bought apple juice has a hydrogen ion concentration of about $h=0.0002$

`h`=0.0002.Determine the pH of the apple juice correct to one decimal place.

Is the apple juice acidic or alkaline?

Acidic

AAlkaline

BAcidic

AAlkaline

BA banana has a pH of about $8.3$8.3.

Solve for $h$

`h`, its hydrogen ion concentration, leaving your answer as an exact value.

Use curve fitting, log modelling, and linear programming techniques

Form and use trigonometric, polynomial, and other non-linear equations

Apply linear programming methods in solving problems

Apply trigonometric methods in solving problems