Logarithms, Part 3

Finally have a little time for a post.

So we know how to solve x2 = 10 by taking the square root of both side of the equation to get x = ±3.162… Note that taking the square root of x2 undoes or reverses the squaring of x.

But what do you do if x is in the exponent and not the base?


You can’t take the xth root since you don’t know what x is. So what to do? From my last post, you saw that log2 10 means “what is the number that I can use as the exponent of 2 so that the answer is 10”. So in the above equation, if I take the log2 of both sides, I get


The left side of this equation is doing two inverse operations on the number 2 – raising 2 to a power then taking its log. In other words, the left side can be seen as saying “what is the number that I can use as the exponent of 2 so that the answer is 2x ?”. Well the answer to that question is x. So the left side is just x and the right side is just a calculation:


Well that’s just dandy! Trouble is, without the internet, how do you find log2 10? I have not seen a calculator with a log2 x button. As mentioned in my last post, calculators usually have buttons to take logs relative to bases 10 and e. Well fortunately, there are lots of properties of logs that can help. The one we can use here is


This means that I can take the log with respect to any base, and the x can be removed as the exponent. So for our problem, let’s take the log10 (the log x key on your calculator) of both sides and see what happens:


Let’s stop here for a moment before I complete the solution. Why is the right side equal to 1? Log10 10 is saying “what power of 10 equals 10?”. The answer is 1 because 101 = 10. On the left side, I used to log property above to bring the x in front of the log. Now log10 2 is just a number. You can use the log x key on your calculator to find that log10 2 = 0.3010 to four decimal places. So now the equation becomes


So 23.3219 = 10. In my next post on logs, I’ll do more equation solving using logs.

Logarithms, Part 2

So what is a logarithm? Let’s first see the notation, then I will explain. When taking the log (short for logarithm which I will use from now on) of a number, you need to know what base is being used. The notation for the log of x is loga x. The a is the base and is usually a specified number. so examples using this notation are log2 10, log10 25, log18 145, loge 7.34. Let’s look at these.

log2 10 is asking the question “What number can I use as the exponent of 2 so that the answer is 10?”. It turns out that 23.321928094887 = 10 so log2 10 = 3.321928094887.

log10 25 is asking the question “What number can I use as the exponent of 10 so that the answer is 25?”. Well, 101.39794 =25 so log10 25 = 1.39794.

Are you getting the picture? What about log18 145? This is asking the question “What number can I use as the exponent of 18 so that the answer is 145?”. 181.72183 = 145 so log18 145 = 1.72183.

Now let’s look at loge 7.34. This shows that the base or the number we are taking the log of does not have to be an integer. The number e, which I have talked about before, is an irrational number, but it still can be used as a base. In fact, it is probably the most used base. Since e1.99334 = 7.34, then it follows that loge 7.34 = 1.99334.

By the way, on most calculators, the log or log x key assumes that the base is 10. On most calculators as well, ln x means loge x. “ln” means “natural log”.

Now loga x and ax are inverses of each other. This means that one undoes the other. So if on your calculator, you find ln 7, then take that number and hit the ex key, you get the original 7 back. This works in reverse as well: Find e7 on your calculator, then hit the ln x key. You will again get the 7 back.

In notation-speak, this inverseness is shown as


In my next post, I will show how logarithms can be used to solve equations.

Logarithms, Part 1

Logarithms confuse many of my students so I thought it is time to explain these. I touched on these before on a post about inverse operations, but let’s add some more detail.

Let’s first define some terms here. Consider the expression x2. Here, x is raised to the power of 2. x is the base and 2 is the exponent, power, order, or index. Lots of different terms for the exponent – I will mostly use the term exponent. So the exponent defines what to do with the base.

Now before I talk about logarithms specifically, I want to review what various kinds of exponents mean. I have talked about this before, but these concepts should be fully understood if logarithms are to make sense to you.

Now x2 means x × x. Positive integer exponents means how many times you multiply the base by itself. So in general, for a positive integer m,

xm = x × x × x × x … where x is listed m times.

The special case of when m = 0 is defined as x0 = 1, no matter how small or how large x is. Now what about negative integers?


So a negative exponent is the same as the positive one except it and its base is in the denominator or vice versa. You can freely move a factor that is a base and its exponent between the numerator and the denominator, as long as you change the sign of the exponent.

What about fractional exponents? Let’s start with fractions where “1” is in the numerator. The denominator in a fraction exponent refers to the root of the number. For example,


The “2” for the square root is usually assumed if it is not there. However, for other roots (like cube roots), the index must be there to indicate the kind of root it is. Other examples:


The numerator in a fractional exponent means the same as if it wasn’t in a fraction. so we can combine these two definitions for more general fractions:


Now we have not covered irrational exponents like x𝜋. The development of these are a bit more complex so I’ll just say “use your calculator”.

Indeed, you can use your calculator to calculate a number raised to a power if it has a key labelled as “yx” or has a key with the “^” symbol on it. I will leave it to you to find out how to use these keys. If you do not have a fancy calculator, there is always the all-knowing internet.

So we have talked before on how to solve equations like x2 = 16 by taking the square root of both sides of the equation. But how do you solve 2x = 16? Notice that x is now in the exponent. That changes everything as you can’t take the xth root of a number on your calculator……………but can you?

In the next post on this topic, I’ll introduce you to logarithms then later, how they are used.

Percentages, Part 3

So now that we know what a percentage is, how is it used? Let’s look at some sample percentage problems.

Melbourne’s Silvan Reservoir has a capacity of 40,446 ML(megaliters). Currently, it is 88% full. How much water is in the reservoir? In other words, what is 88% of 40,446?

Whenever you see or interpret a problem where you need to take a percentage “of” something, equate the word “of” with “multiply”. So to take 88% of 40,446, we multiply 40, 446 by 88%. If you do this on a calculator, you need to use the decimal equivalent of 88% which is 0.88 (see my previous post). On some calculators, there is a percent key. On these, you can type 88 which the calculator will interpret as 0.88 when you use that key. Regardless of which calculator you use, when you multiply 40, 446 by 88%, you should get 35592.48 ML.

If you do not live in Melbourne, the above problem does not interest you much. As money is of interest to most everyone, let’s look at some typical money related percentage problems.

Since DavidTheMathsTutor has effectively educated the masses on how to use percentages, calculators with percentage keys are no longer in demand. So a store has discounted the normal $24.95 price of these calculators by 30%. What is the new price?

This is a two-step problem: first find what the amount of the discount is, then subtract it from the original price. The amount of the discount is 30% × 24.95 = 0.3 × 24.95 = $7.48 (rounded to 2 decimals as we are talking about money). So the store is taking $7.48 off each calculator. So the new price is 24.95 – 7.48 = $17.47.

On the other hand, again because of DavidThe MathsTutor, there is a big demand on fancy calculators that do all sorts of mathematical things like graph equations. So the store decides to markup the normal price of these calculators by 25% to make up for the loss of the percentage calculators. The normal price of these are $149.50 each. What is the new price?

Again, a two-step problem, but this time you are find the amount of the price increase, then add it to the original price since this is a markup. The amount of markup is 0.25 × 149.50 = $37.38. So the store is increasing the price by $37.38, so the new price is 149.50 + 37.38 = $186.88.

Many times, you need to calculate the original price. For example, you are looking to buy a car. The sticker says $24,500. The salesperson says that’s a good price because they are only making a 5% profit on it. What is the cost of the car to the store? This is a reverse markup problem: what price plus 5% of that price is $24,500?

In equation form, that is the equivalent maths sentence, this is

x + (0.05 × x) = 24,500.

If you remember my posts on equations, factoring, and the distributive property, this is solved by factoring the left side and then dividing both sides by the number that results from the factoring:


If you knew the cost to the store was $23,333.33 and knew that they marked up that cost by 5%, if you calculated the new price as we did before, you would get $24,500.

Sometimes you need to calculate the actual percentage. What if your salary went up from 88,000 to 99,000? What percentage pay rise is this (so you can brag to your friends)?

The amount of pay increased by 99,000 – 88,000 = 11,000, so we want to know what percentage of 88,000 is 11,000. Note that you always work with the original price or amount when working out a percentage. So the equivalent maths sentence is

88,000 × x% = 11,000

Dividing both sides by 88,000 gives


You must be very good at your job!

Percentages, Part 2

So how do you convert percentages to fractions and decimals and vice versa? This post will show examples of each.

  1. Convert a percentage to a fraction:

This one is easy as if you remember, a percentage is already a fraction where the numerator is displayed and the denominator is 100. So you just create the fraction and simplify it (see my posts on fractions):


2. Convert a percentage to a decimal:

This one is just a matter of moving the decimal point, two places to the left. Keep in mind that the decimal point will not usually show at the end of integer percentage, but you can assume it to be at the end of the number:

37% = 37.% = 0.37

18.5% = 0.185

112% = 1.12

0.15% = 0.0015

Any 0’s at the end of the decimal, can be left off:

40% = 0.40 = 0.4

3. Convert a decimal to a percentage:

This is just the opposite of of the above: you just move the decimal point two places to the right, then add the % symbol:

0.25 = 25% (if an integer results, you can leave the decimal point off)

0.2786 = 27.86%

0.002 = 0.2%

2.345 = 234.5%

4. Convert a fraction to a percentage:

Here you multiply by 100/1, simplify, then multiply numerators together and denominator together. It is advisable to simplify before multiplying:


Sometimes though, not as much cancels and you will need to do some division in the end (long or short – see my post on long division):


In my next post, I will show how to do some of the more common problems using percentages.

Percentages, Part 1

I am confident that the chance of you liking this post is 100%. But what does 100% mean? I will talk about this in the next few posts.

First let’s review what a fraction is. The fraction


means that I have 3 pieces (the numerator) of some whole thing that was divided into 4 pieces (the denominator). So the denominator sets the size of the pieces (the bigger the denominator, the smaller the pieces) and the numerator sets how many pieces.

Being creatures that have 10 fingers, we naturally migrate to things that are powers of 10. Way back in the Roman empire days (I remember them fondly), the Romans frequently used fractions that had a denominator of 100. In fact, the word “percentage” comes from the latin (Roman) per centum which means “by the hundred”. That has continued to today, as a percentage is really a fraction where the denominator is fixed at 100. What you see in a percentage is the numerator:


So a percentage like 37% means that if I divide something into 100 equal pieces, I have 37 of these pieces. Percentages are also used to indicate a probability. My first sentence in this post used a percentage in this way. If you remember, a probability of an event is a fraction. The probability of event A is expressed as P(A):


So 100% of you liking this post means that if I take 100 random people who have read this post, 100 of them will like this post. This means all will like this post as that is what 100% means: the whole:


Now percentages, fractions, and decimals are all ways to express a part of something. In my next post, I will show how to convert percentages to fractions and decimals and vice versa.

Factors of Polynomials, Part 4

So we are on a journey of how to factor a polynomial like


and the motivation to do this is so we can easily solve equations where the polynomial equals 0. For example, if we have


where the left side is a factored form of a polynomial, you can quickly see that x = 7, -3 or 2 will solve the equation since these numbers make one of the factors 0.

The last post reviewed some of the tools you can use if you have a quadratic (highest power of x is 2) polynomial. But what can you do if you have a polynomial like the one at he beginning of this post? Well there are two theorems regarding polynomials that will help here: The Remainder Theorem and the Rational Root Theorem.

The modified Rational Root Theorem says that if a polynomial has an integer root (that is a value of x that makes the polynomial equal to 0), then that value of x must be an integer root of the constant term (the number without any x‘s). This is a modified version of the Rational Root Theorem because I am restricting the coefficient in front of the highest power of x to be 1.

So for our polynomial, the constant term is 30. So the possible roots are ±1, ±2, ±3, ±5, ±6, ±10, or ±15 as these are all factors of 30. So how do we check if any of these numbers are a root? Along comes an application of the Remainder Theorem: If a is a root of a polynomial (that is (xa) is a factor of the polynomial), then the polynomial evaluated at a equals 0. This is really a specific application of the more general Remainder theorem, but this is all we need.

What this means is that if I want to test if a number is a root of a polynomial, all I have to do is evaluate (replace the x‘s) with that number and see if I get 0. If I do, the the number is a root, if not, then it’s not.

So let’s check if 1 is a root, that is check if (x – 1) is a factor:


So (x – 1) is not a factor. Let’s try -2:


Great! That means (x + 2) is a factor. Now from here we can do two things: either divide our original polynomial by the newly discovered factor (x + 2) or we can continue using the Remainder Theorem to test the other integer roots.

Polynomial division is not hard but It’s just as easy to continue using the Remainder Theorem. If you check x = 3 and x = 5, you will see that the polynomial does equal 0. So the compete factorisation of our polynomial is:


Don’t you feel mathematically powerful!!

Factors of Polynomials, Part 3

As the order of the polynomial (the highest power of x) increases, it usually gets harder to factor. In my last post on this topic, I will cover a way to reduce the order by one for each iteration of the process. If you can get the polynomial to a degree 2, there are many ways to factor these.

A polynomial of degree 2 is called a quadratic. I covered factoring quadratics or solving quadratic equations (equations where the quadratic is set equal to 0) in several posts before. Please review these but I will just remind you of them here.

A quadratic is a polynomial of the form \[{ax}^{2}\hspace{0.33em}{+}\hspace{0.33em}{bx}\hspace{0.33em}{+}\hspace{0.33em}{c}\hspace{0.33em}\]

where a, b, and c are some numbers.

Now for this set of posts, I am restricting a to be 1. So we would like to factor the quadratic to look like \[{(}{x}\hspace{0.33em}{+}\hspace{0.33em}{a}{)(}{x}\hspace{0.33em}{+}\hspace{0.33em}{b}{)}\]

Basically, the method is to do a reverse distributive property (please see my posts on this). Let’s do an example. Let’s factor \[{x}^{2}\hspace{0.33em}{+}\hspace{0.33em}{2}{x}\hspace{0.33em}{-}\hspace{0.33em}{24}\hspace{0.33em}\]

We need to find two numbers, a and b, so that they multiply to equal -24 and add or subtract to equal +2, the coefficient in front of the x. 8 and 3 do not work as they do not add to equal 2. However, 6 and 4 look like contenders. The signs of 6 and 4 must be such that they add to equal +2 but multiply to equal -24. Looks like +6 and -4 work so \[{x}^{2}\hspace{0.33em}{+}\hspace{0.33em}{2}{x}\hspace{0.33em}{-}{24}\hspace{0.33em}{=}\hspace{0.33em}{(}{x}\hspace{0.33em}{+}\hspace{0.33em}{6}{)(}{x}\hspace{0.33em}{-}\hspace{0.33em}{4}{)}\hspace{0.33em}\]

Another method you can use is to find the zeroes of the quadratic directly instead of factoring. This method is the quadratic formula. Please see my prior post on this.

The quadratic formula is \[{x}\hspace{0.33em}{=}\hspace{0.33em}\frac{{-}{b}\hspace{0.33em}\pm\hspace{0.33em}\sqrt{{b}^{2}\hspace{0.33em}{-}\hspace{0.33em}{4}{ac}}}{2a}\]

where a, b, and c are the coefficients in the general form of a quadratic.

From the example I just factored, we can see that x = -6 and x = 4 are zeroes of the quadratic. I could find these directly using the quadratic formula:


So looks like we have a few tools available to factor quadratics. But what can we do if the order of the polynomial is higher than 2? I will cover a method to do this in my next post.

Factors of Polynomials, Part 2

So we are talking about polynomials and how to factor them. We want to factor a polynomial in order to easily find its zeroes, that is, the values of x that make the polynomial equal to 0.

First a definition: the order or degree of a polynomial, is the highest power of x in the polynomial. So x² -3x + 7 is a polynomial of degree 2.

The first method to discuss is the easiest to apply. If the polynomial has an x in each term, you can factor that out. This will show that x itself is a factor of the polynomial. Let’s do an example.



Notice that there is at least one x in each term. We can “undistribute” this x and make it a factor. Please review my posts on the Distributive Property.


Since x by itself is a factor of the polynomial, 0 itself is a zero of the polynomial. So now we need to complete the factoring process by factoring the stuff in the brackets.

The expression left in the brackets is called a quadratic which means it’s a polynomial of degree 2. Now I’ve discussed methods before on how to factor or find the zeroes of a quadratic. I will review these in my next post.

Factors of Polynomials, Part 1

Many problems in engineering and science involve finding the zeroes of a polynomial. This means finding the values of x such that the polynomial is zero. But let’s review what a polynomial is.

A polynomial is anything that can be put in the form:


where n is a positive integer and the a‘s in front of the x‘s are any real numbers. The numbers in front of the x‘s are called the coefficients. For this post I will only be looking at polynomials with integer (positive or negative) coefficients and polynomials where the first coefficient is 1.

Some examples of polynomials are:


Notice that all the decreasing powers of x do not have to be present. Also, note that numbers by themselves are polynomials as n in this case is 0 and anything to the 0 power is 1.

Now to find the zeroes of these things often requires us to factor the polynomial. That is, change the form of the polynomial t0 several things multiplied together:


And we want to do this because of the Null Factor Law. Please see my post about this law but it means that once a polynomial is factored, the values of x that make each factor 0, make the whole polynomial 0.

As an example, it is not obvious what values of x make x² -11x + 30 equal to 0. But if you knew that this polynomial is also equal to (x – 5)(x – 6), then you can immediately see that x = 5 and x = 6 are the zeroes of these factors and are therefore the zeroes of the polynomial.

Now it is not always easy to factor polynomials, but the next few posts will talk about some methods to help do this.