A-level maths or advanced level mathematics is the maths you do in your higher education, we have established that multiple times in some of our previous posts. It is the stepping stone to your work life.
That is because after getting an A level certificate or diploma, you will be able to apply to most jobs in various industries or even ask for a raise. For instance, after concluding a maths A level course you are eligible for promotion in most government jobs.
They value your maths skills because it is crucial in the workplace, skill sets you gain from learning maths include logical thinking, critical thinking, analytic, and problem-solving. Hence, A level maths serves as a tool for gaining these crucial skills.
And one of maths A-level topics that are very useful in acquiring those skills is logarithm and indices. An exciting topic and a very interesting one, if you have an interest in solving maths problems anyway, and if you want to pass that maths A level course, logarithm is a fundamental topic in further or A level maths.
You have to understand its concepts and learn how to solve logarithm problems before delving deeper into more maths lessons.
Lessons that will expand as you progress, into more complex maths equations, with logarithm and the other core topics forming the basis of the solutions for the maths problems. The other topics include calculus and mechanics, you will come across more while reading math textbooks or when studying maths A level past papers.
Logarithm and indices deal with solving exponential equations and problems, it has a set of rules known as the laws of logarithm that enable you to solve recurring maths problems or ones that have a certain format.
You can read more about them below, we have compiled some of the rules and a brief description of each, and how you can use them to solve some simple maths questions to let you have an idea what to expect once you start your A level maths program in Nigeria.
There are some logarithm special cases we need to deal with before we can get on the way. The exponent is the opposite of a logarithm, it is usually written with an e followed immediately by an x as a superscript. Like so;
The exponential function or expression is written as;
y = ex
So that e the exponent form is a constant in this whole maths expression, and it never changes, it is actually representing a specific real number that has lots of decimal places after it, just like our revered Pi, π.
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To solve logarithmic functions we use something known as the natural log written with the notation “In”. It is the inverse of an exponent form. If you remember your differentiation and integration lessons you will remember that if you differentiate a function, the derivative of the function is always written as dy/dx. And it is always equal to some number.
Well, the derivative of the exponent form e is equal to itself, that is why it is a constant. You will learn more about it if you become good at differentiation and integration. Regardless, the expression is written as;
dy/dx = ex
Here is an example of a logarithm question that we can solve using natural logs before we commence with the laws of logarithm.
Natural Logs Example Question
To solve for x in the exponential function below,
0.5 = ex
You begin by deciphering the 'e' in the equation. To be totally sincere this is nothing like the algebraic functions you were used to, mostly because we know the 'e' or (exponential form) is a constant. So we can’t just throw around the letters and numbers, rearranging them until it won’t go any further anymore.
There is a simple solution to this, and it is by adding the natural log notation “In” to the equation. Remember that it is the inverse of the exponent function, so when you add it to both sides of the equation above, it will cancel out the exponent function, it is the easiest way to get rid of the 'e' in the equation to find the value of x.
It goes like this;
In (0.5) = In (ex)
The inverse of any number is simply one divided by that number, 1/number. So when you multiply the inverse of a certain number by that number they cancel out themselves.
That is exactly what is happening here, In is the same as saying 1/e, so that,
1/e x e = 1
Therefore, our answer after the exponent form is taken cared of is x equal to the natural log of 0.5. Written as;
In (0.5) = x
This is our solution, we have found x, and more needn’t be done. But for the sake of curiosity, or you really must solve x to give you a figure you can use, then the mathematical constant e is approximately equal to 2.71828, therefore making In the natural log 1/2.71828 in the equation.
If you punch in the equation in a calculator starting with the In notation, it will reveal the true answer.
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How To Solve Logarithm
Logarithm was invented to simplify big numbers in maths, it deals with powers, i.e. when a number is raised to the power of something. When you have such a scenario, where the power is involved in an equation, then it is possible for logarithm to play a role there.
When you say log something, that something being a number, of course, it means the log of that number is equal to the power the number is raised to. For example, log 100 is equal to two.
All this, however, is assuming the base of the log is 10. So, it’s actually logbase10 where the base 10 is not written, unless it is a different number, like 4, 5, 6, or 7. A Logarithm equation looks like this;
Log 100 = 2
Take note, this is the same as saying 102 is 100. Also;
Log5 625 = 4
Means, 54 = 625. It is quite simple when you understand this basic way to solve logarithms. In the second example, the base is often represented by an “a”, written loga. Where “a” is any number.
Laws Of Logarithm
Like we stated earlier the rules known as the laws of logarithm are there to make sure we stay on course and that we solve maths problems that have specific forms involving logs.
These laws can be applied to any base system as long as it is the same all through. Like a base 5 such as the one we used in the example above, the laws are applicable if the whole expression has base 5 as its base.
Without further ado, here they go.
The First Law Of Logarithm
loga x + loga y = loga (xy)
The first law states that adding the logarithms of numbers is equal to the log of the numbers multiplied by each other. Meaning, log 2 + log 3 can be simplified into log (2x3).
The Second Law Of Logarithm
loga x - loga y = loga (x/y)
The second law deals with subtraction in logarithm expressions, it states that subtracting the logs of two numbers or integers is equivalent to dividing the two numbers and taking the logarithm of whatever result you got from the division.
And of course, they have to have the same base for the rule to work, same goes for the first and third law.
The Third Law Of Logarithm
loga xn = nloga x
The third law is quite interesting, if a number is raised to power, and it says you have to find its log. Then you should know that the log of that number that is raised to power of something, is the same thing if you remove the power from the number and multiply the whole log expression by it.
For example, log 75 is the same thing as saying 5Log7.
Rounding Up The Laws Of Logarithm
The laws of logarithm as you can see all seem pretty easy but in reality, they could be a lot lot harder to solve, because you could get a logarithm question that needs a combination of two or all of them before it can be solved. And while most maths functions are equations, that is with an equal sign, some of them are expressions, meaning they don’t have an equal sign.
And many logarithm questions are all about simplifying the expression than really adding up or subtracting anything from an equation. So when solving any A level maths equation you ought to be mindful of what it is you are solving.
Logarithm and indices creep their way into many maths problems (especially calculus) which is why they are fundamental when learning A level maths. As insignificant as the equations and expressions and laws may seem, they actually form the nuts and bolts of many unrelated maths problems.
Maths A level is a rather wide academic course, and one of the most fundamental topics r concepts you will learn about during the course of your study is differentiation and integration, take a peek here.