By Evan Ma
(Math and Physics tutor at The Edge Learning Center)
where , a strictly positive real number, is called the base and is the power. For example, we all know the following by heart:
and so on. The logarithmic function does it backwards – given a number as the argument, it yields the power subject to a specified base, and therefore it is defined as follows.
Therefore, it is clear that
and so on. Let’s look at other examples involving other bases:
Let’s complete the following questions as practice: (a) , (b) , (c) , (d) . The answers can be found at the end of this blog.
From the definition of logarithm, we can also see that if we raise the base by the base-logarithm of a real number , we will get back the number , viz
You may try to verify this by using your calculator for different valid bases. This identity is important when we try to solve logarithmic equations, examples of which will be given alter.
Now, let’s take a look at the rules of the logarithm. Similar to the definition of the logarithm, we use the laws of indices to derive the rules:
1. Logarithm of a product is sum of the logarithms
Proof: Say and , therefore
2. Base- logarithm of Mr is times
Proof: Say , therefore
3. Logarithm of a quotient is the difference of logarithms
Proof: Say and , therefore
4. The final rule is called the change-of-base formula. We shall use an example as a tool to derive the formula. Let’s say we want to know what power must 2 be raised in order to get 50. As 50 is not an integer power of 2, the answer is not very obvious. Hence, we write
and therefore. In order to find , say we apply base-10 logarithm to both sides of the equation as our calculator may not have the base-2 logarithm key:
Corrected to 4 significant figures, the value of is approximately 5.644. In fact, you can verify your answer by raising 2 to the power of 5.644 to see that the answer is approximately 50.
To generalize the problem, let’s say we have to find such that and base- logarithm is not at our disposal, we can therefore instead use base- logarithm according to the equation
By way of example, let’s consider the following. Suppose an amount of $10,000 is deposited at an interest of 2.5% per annum, compounded annually. How long must the money be kept in the account for it to grow to $50,000?
To answer the question, we are basically trying to find n such that
As my simple scientific calculator does not allow me to specify a base of 1.025 for the logarithmic key, I will have to rely on the change-of-base formula and use base-10, and hence
Since interest is compounded annually, it will take 66 years to grow at least 5 times.
Having explored rules of logarithms, we introduce the natural logarithmic function, or . It is base-, where is the eminent irrational number ≈2.71828 and whose significance in sciences and mathematics cannot be emphasized enough. Hence, base-logarithm is defined as follows.
Next, let us take a look at two examples where common mistakes are made in solving equations of logarithms. See if you can identify the mistake.
Where is the mistake? Look closer. Of course, in the second step, one cannot “split” the logarithm across the addition sign. Remember, you can only “split” a logarithm into a sum if the logarithm is applied to a product, not a sum. The correct steps are therefore, as follows:
Now, let’s look at the following example, and see if you can identify the mistake:
Where is the mistake? Yes, it is in the second step – a quotient of logarithms is of course not the logarithm of the quotient. Rather, we can use the change-of-base formula to simplify the first step, as follows:
Finally, we will illustrate how to use rules of logarithm to solve the following:
You may notice that the unknown appears as the base and a variable in the argument of the logarithm. How can we solve for ? The method still depends on applying the rules of logarithm consistently. First, we raise both sides as powers of the base , and hence
You may recall from the definition of logarithm that the left hand side will just become the argument of the logarithm, and therefore
By re-arranging the above equation, we need to solve
Noticing that this resembles a quadratic in , we solve for as follows:
Now, where is the negative root? Since is also the base of the logarithm, the negative root is therefore rejected as a solution. Hence, is the final answer.
By reviewing the above examples, you may see that solving a seemingly difficult logarithmic equation is not hard at all – simply apply the rules of logarithm consistently and the correct solution can be obtained.
Answers to questions: (a) 2 (b) 3 (c) -3 (d) -2.