Understanding Logarithms

By Evan Ma

(Math and Physics tutor at The Edge Learning Center)

log-a-rhythms logarithmic function

Log-a-rhythms (Photo taken from Pinterest)

The logarithmic function seems to confound many students studying the IB Math SL program. In order to understand it properly, let us define

ar-m

where a, a strictly positive real number, is called the base and r is the power. For example, we all know the following by heart:

10-3-0-001

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.

If

ar-m-a-0

then

r-loga-m

Therefore, it is clear that

log-10-0-0001

and so on. Let’s look at other examples involving other bases:

23-8-log2

Let’s complete the following questions as practice: (a)log10010000 , (b)log5125 , (c) log91729  , (d)log1749 . 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-alogarithm of a real number m, we will get back the number m, viz

alogamm

logarithmic-keys

(Image taken from Wikimedia Commons)

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 apm and aqn, therefore

mn-apaq

2. Base-a logarithm of Mr is r times logam 

Proof: Say apm, therefore

mr-apr
 3. Logarithm of a quotient is the difference of logarithms

Proof: Say apm and aqn, therefore

mn-apaq

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

2r50

and thereforerlog250. In order to find r, say we apply base-10 logarithm to both sides of the equation as our calculator may not have the base-2 logarithm key:

log102rlog1050

Corrected to 4 significant figures, the value of r 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 r such thatarm and base-a logarithm is not at our disposal, we can therefore instead use base-b logarithm according to the equation  

rlogblogba

and therefore

logamlogbmlogba

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

1000010-025

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

nlog105

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 inx . It is base-e, wheree is the eminent irrational number e≈2.71828 and whose significance in sciences and mathematics cannot be emphasized enough. Hence, base-elogarithm is defined as follows.

If

erm

then

rinm

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.   log2x224

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:

log2x224

Now, let’s look at the following example, and see if you can identify the mistake:

in27x-1in3-2

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:

in27x-1in3

Finally, we will illustrate how to use rules of logarithm to solve the following:

logx6x2-94

You may notice that the unknown x appears as the base and a variable in the argument of the logarithm. How can we solve for x? The method still depends on applying the rules of logarithm consistently. First, we raise both sides as powers of the base x, and hence

xlogx6x2-9

You may recall from the definition of logarithm that the left hand side will just become the argument of the logarithm, and therefore

6x2-9x4

By re-arranging the above equation, we need to solve

x4-6x29

Noticing that this resembles a quadratic in x2, we solve for x as follows:

x2-32

Now, where is the negative root? Since x is also the base of the logarithm, the negative root is therefore rejected as a solution. Hence, x3 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.