Logarithm: Using The Shift And Push Technique To Solve Logarithm Equations

Introduction

Welcome back to another Secondary Math blog post!

Do you find logarithm equations complicated?

In today’s blog post, I will teach you how to solve Logarithm equations using the Shift and Push Technique by analysing a question from the 2018 Fuhua Secondary School (FHSS) S3E SA1 Examination Paper.

Alternatively, you can watch my explainer video on this question here.

Let’s Take A Look At This Logarithm Question

Source: Fuhua Secondary School — 2018 S3E SA1 Examination Paper [Q10(i)]

Thought Process

💡 What Is A Logarithm Function? 💡

A logarithm function, or log function in short, can be rewritten as logₐx.

The a term is called the base term, while x can represent any expression or certain constant value.

🔎 How Can A Logarithm Be Well-Defined? 🔎

1️⃣ The base term should be positive (a > 0).
2️⃣ The base term cannot take up the value of 1 (a ≠ 1).
3️⃣ The x must always be positive (x >0).

If these three conditions are met, the log term is well-defined.

Here are some examples of well-defined logarithm functions:

But what do these Logarithm Functions really tell us?

To answer that, we will call upon the Exponential Function, which can be represented in the form of y=a^x.

Do you notice a pattern when you compare each of the corresponding examples?

The base terms of 2, 3, and 4, and the numbers 8, 9, and 16 repeat in both functions.

However, there is one number that did not repeat on both sides — the powers of the exponential function.

This is exactly what logarithm functions mean. They are the power that the base term will be raised to. Logarithm and exponential functions are related to each other.

In our first example, 2³ gives us 8. Therefore, log₂8 will give us 3.

In the second line, log₃9 will give us 2.

What about log₄16? We will get 2!

Let Us Solve Question 10(i)

Now that you have a better understanding of what the logarithm function represents, let us go back to Question 10(i) to solve the logarithm equation.

Source: Fuhua Secondary School — 2018 S3E SA1 Examination Paper [Q10(i)]

To solve this equation, we’ll be required to remove the log function.

How do we do that?

Let us use log₂8 as an example. We know that this will be equal to 3 and we can rewrite this as 8=2³.

So how do we go about it?

First, recall that the 2 in our logarithm function is known as the base term, while the number 3 is known as the power term.

To remove logarithm functions, we can use the Shift and Push Technique.

🔃 What Is The Shift and Push Technique? 🔃

It is a technique used to express any logarithm function in its exponential form.

How does it work?

Notice that when we shifted 2 to the other side, it went into the base position. At the same time, it pushes whatever is on the right initially — in this case the number 3 — into the power position.

Now that you have the idea of how to remove logarithm from an equation, let’s proceed to solve the question.

Source: Fuhua Secondary School — 2018 S3E SA1 Examination Paper [Q10(i)]

In the given equation, x plays the role of our base while 2 on the right-hand side plays the role of our power term.

We shall remove log from this equation by using the Shift and Push Technique, which removes log to give us this equation:

72 – x² = x²

Now, we can proceed to solve this equation.

Because this is a logarithm question, there is one last step that we need to do — to check if our final answer satisfies the three laws that we discussed earlier.

Note that x plays the role of the base in the equation. Hence, x cannot take up negative values.

This means that we have to reject -6, simply because x must be positive.

Therefore, there is only one final solution.

Final Answer

x = 6

Conclusion

I hope that after reading this blog post, you have gained a better understanding of the relationship between the logarithm and exponential functions.

Remember to use the Shift and Push Technique to convert any logarithm function to an exponential function!

Stay tuned for more Secondary Math tips!