Rules of Logarithms | Don't Memorise

Summary

The video introduces the three basic rules of logarithms frequently used: logarithmic addition, subtraction, and power identities. The logarithmic addition identity states that the log of a product is equal to the sum of logs. The subtraction identity explains that the log of a quotient is the difference of logs, while the power identity notes that the log of a power can be expressed as the power times the log. The video warns of common student mistakes and demonstrates application through an example, solving for log base 2 of 24 using the given log of 6 and known properties.

Highlights

• The logarithmic addition identity involves the sum of logs when multiplying inside the log. ➕
• Subtraction identity requires the same base for logs, leading to the difference of logs when dividing. ➖
• The power identity simplifies logs involving exponents by multiplying the exponent with the log. 🔢
• A common error involves misunderstanding log identities, such as misapplying multiplication within the log. 🚫
• Demonstrated problem-solving with log base 2 of 24 using known values and the addition rule. ✅

Key Takeaways

• Logarithmic addition: log(xy) = log(x) + log(y). 📚
• Logarithmic subtraction: log(x/y) = log(x) - log(y). ➗
• Logarithmic power: log(a^n) = n * log(a). 📈
• Common mistake: log(x * y) ≠ log(x) * log(y). ❌
• Practical application: Break down complex logs using known values and identities. 🔍

Overview

Welcome to a thrilling mathematical adventure into the world of logarithms! The video starts by laying down the foundational rules: first up is the addition identity, where multiplying within the log translates to addition outside. Next, highlights the subtraction identity, where division is mirrored as subtraction between logs.

Another intriguing identity is introduced: the power property of logs, which unfolds the mystery of exponents by converting the exponent into a multiplication factor with the log. This video carefully navigates common pitfalls students face, like equating log multiplication incorrectly.

The video isn't just about theory! It delves into a pragmatic example, calculating log base 2 of 24 using provided values and identities. By breaking 24 into a product (4 times 6), and employing known values, the solution beautifully emerges, right on point, showcasing the power of logarithmic identities.

Chapters

• 00:00 - 00:30: Introduction to Logarithms This chapter introduces the concept of logarithms and focuses on important rules of logarithms. It highlights the first key rule, known as the logarithmic addition identity, which states that the logarithm of a product 'xy' to the base 'b' is equal to the sum of the logarithms of 'x' and 'y' to the base 'b'.
• 00:30 - 01:00: First Two Logarithmic Rules The chapter titled 'First Two Logarithmic Rules' introduces two fundamental logarithmic identities. It emphasizes the importance of having the same base for these rules to be valid. The first rule discussed is the product rule, which states that the logarithm of a product is the sum of the logarithms of the factors, given the base is the same for all logarithms involved. The second rule is the quotient rule, which states that the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator, again requiring the same base for validity.
• 01:00 - 01:30: The Third Logarithmic Rule The chapter introduces the third logarithmic rule known as the logarithmic subtraction identity. This rule focuses on the situation where the logarithm of 'a' raised to 'n' to the base 'b' equals 'n' times the logarithm of 'a' to the base 'b'. The power of the argument plays a significant role here, and it's emphasized as one of the three crucial rules in logarithms. Additionally, the chapter touches on common mistakes students might encounter while applying this rule.
• 01:30 - 02:00: Common Mistakes with Logarithmic Rules This chapter addresses common misconceptions with logarithmic rules. A key point discussed is the incorrect belief that the logarithm of a product (log x * log y) equals the sum of the logarithms (log x + log y), which is not true. Similarly, it covers the misunderstanding that the logarithm of a division (log x / log y) equals the difference between the logarithms (log x - log y), which is also incorrect. The chapter emphasizes that the product or division should appear as the argument inside the logarithm function, not outside. An example is provided to illustrate these principles.
• 02:00 - 02:30: Example Problem Setup The chapter begins by posing a mathematical problem: finding the logarithm of 24 to the base 2, without using a calculator. It acknowledges that most calculators offer log base 10 and natural log but not base 2. Additionally, it provides a clue by stating that the log of 6 to the base 2 is known to be 2.58496. The problem is set up to explore methods of calculating this logarithm using the given information.
• 02:30 - 03:00: Applying the First Rule The chapter introduces the concept of finding the value of a logarithm, specifically 'log 24 to the base 2', by breaking down the number 24 into a product of two numbers. It suggests writing 24 as '4 times 6' rather than '2 times 12' to simplify the problem, hinting that the reasoning behind this approach will be explained later.
• 03:00 - 04:00: Using the Third Rule In this chapter titled 'Using the Third Rule', the rule of logarithms is discussed and applied to solve a problem. The specific rule explained is that the logarithm of a product can be expressed as the sum of the logarithms of its factors: 'log xy = log x + log y'. An example is given where the logarithm of 4 to the base 2 is added to the logarithm of 6 to the base 2 to demonstrate this rule. The chapter underscores that understanding logarithmic identities aids in simplifying and solving expressions, in this case explaining why 'log base 2 of 4' equals 2, since 2 raised to the power of 2 gives 4.

Rules of Logarithms | Don't Memorise Transcription

• 00:00 - 00:30 There are many rules of logarithms, and out of those, there are three which are frequently used Here's the first one. It tells us about the log of 'xy' to the base 'b'. You see that there's a product inside. So the first rule states that this equals log of 'x' to the base 'b' plus log of 'y' to the base 'b'. Yes, this is the first rule and it's called the logarithmic addition identity. When two logs with the same base are added
• 00:30 - 01:00 we write the argument as the product of the two arguments. The most important thing we should notice here is the base. The base has to be the same. Only then will this rule hold true. The next rule talks about log of 'x' over 'y' to the base 'b'. You can probably guess what this will be equal to. Yes, it will be equal to log of 'x' to the base 'b' minus log of 'y' to the base 'b'. That's the second rule and again the bases are the same.
• 01:00 - 01:30 This is referred to as the logarithmic subtraction identity. The third rule is quite interesting! It tells us about the log of 'a' raise to 'n' to the base 'b'. Here the base is 'b' and the argument is 'a' raise to 'n'. It equals 'n' times log of 'a' to the base 'b'. The power of the argument is written here. So these are the three most important rules in logarithms But there are a few common mistakes that students make
• 01:30 - 02:00 while remembering these rules. Log 'x multiplied by log 'y' is not equal to log of 'x' plus 'y'. Log of 'x' times 'y' is equal to log of 'x' plus log of 'y'. And similarly log of 'x' divided by log of 'y' is not equal to log of 'x' minus 'y'? Remember that the product or the division is written as the argument, not here. Now where do these rules help us. Let's look at an example.
• 02:00 - 02:30 We have been asked to find the log of 24 to the base 2. Now most calculators provide the answers for log to the base 10 and the natural log. So assume that we need to find this value without the calculator. But we are given some additional information along with it. The 'log of 6 to the base 2' is equal to 2.58496. How do we approach this problem? We are given the value of 'log 6 to the base 2'.
• 02:30 - 03:00 And we need to find the value of 'log 24 to the base 2.' Give it a try. The trick lies in writing 24 as a product of two numbers. This can be written as 'log of 4 times 6 to the base 2'. The argument '24 is written as 4 times 6'. Now why did we write it as '4 times 6' and not '2 times 12'? You'll know soon!
• 03:00 - 03:30 Let's try applying the first rule here. We can write it as 'log of 4 to the base 2' plus 'log of 6 to the base 2.' Log of 'xy' is log of 'x' plus log of 'y'. It's the same concept we applied here. We have been given the value of this term, but what about this one? Do we need the calculator for this? This just asks us, '2 raise to what will give us 4?' And the answer is 2. There was another way in which we could have solved for the first term.
• 03:30 - 04:00 It can be written as 'log of 2 squared to the base 2'. Applying the third rule we can write this as '2 times log of 2 to the base 2'. And we know that log of 'b' to the base 'b' equals 1. And 2 times 1 is 2. No matter what rule or property you apply, you will get the same answer. Anyway, coming back to our problem. This is given to us as '2.58496'.
• 04:00 - 04:30 Adding these terms gives us '4.58496'. Let's verify our answer using an online calculator. The base is 2 and the argument is 24. If we calculate this, we get '4.58496'. Our answer is correct. Before we move on to the other examples, we will prove these three rules in the next few videos.