New Zealand
Level 8 - NCEA Level 3

# Applications of Logarithmic Functions (y=klogx+c)

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.

#### Exploration

• 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$36n total possible combinations for a number plate of length $n$n, and so we have the equation

$x=36^n$x=36n

We can make $n$n the subject by rewriting the equation in logarithmic form:

$n=\log_{36}x$n=log36x

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 $=$= $\log_{36}x$log36​x (Writing down the equation) $n$n $=$= $\log_{36}2000000$log36​2000000 (Substituting) $n$n $\approx$≈ $4.0487...$4.0487... (Evaluating) $n$n $=$= $5$5 (Rounding up)
Since we can only have a whole number of characters, we want to round our number 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=log36​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 $=$= $\log_{36}64000000$log36​64000000 (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.

#### Practice question

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=log10h, where $h$h is the substance’s hydrogen ion concentration.

1. 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.

2. Is the apple juice acidic or alkaline?

Acidic

A

Alkaline

B

Acidic

A

Alkaline

B
3. A 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.

### Outcomes

#### M8-4

Use curve fitting, log modelling, and linear programming techniques

#### M8-7

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

#### 91574

Apply linear programming methods in solving problems

#### 91575

Apply trigonometric methods in solving problems