Simplify And Evaluate Logarithmic Expressions A Step By Step Guide

Hey guys! Today, we're diving into the exciting world of logarithms. We'll be tackling the expression log₂ 20 + log₂ 5 - log₂ 4, and I'll show you how to simplify and evaluate it like a pro. Logarithms might seem intimidating at first, but trust me, with a few key properties and a little practice, you'll be solving these problems in no time. Our goal is to break down this expression, making it easier to understand and calculate. So, grab your thinking caps, and let's get started!

Understanding the Basics of Logarithms

Before we jump into simplifying our expression, let's quickly review the fundamental concepts of logarithms. Think of a logarithm as the inverse operation of exponentiation. In simpler terms, if we have an equation like 2³ = 8, the logarithm answers the question: "To what power must we raise 2 to get 8?" The answer, of course, is 3. We write this as log₂ 8 = 3. The subscript 2 is the base of the logarithm, and it tells us what number is being raised to a power. The number 8 is the argument of the logarithm, which is the value we want to obtain by raising the base to a certain power.

Now, let's talk about the properties of logarithms that will be our best friends in simplifying expressions. There are three main properties we'll be using today:

1. Product Rule: logₐ (xy) = logₐ x + logₐ y. This rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. In simpler terms, if you're taking the log of two things multiplied together, you can split it into the log of the first thing plus the log of the second thing.
2. Quotient Rule: logₐ (x/y) = logₐ x - logₐ y. This rule is similar to the product rule, but for division. The logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. So, if you're taking the log of something divided by something else, you can split it into the log of the top minus the log of the bottom.
3. Power Rule: logₐ (xⁿ) = n logₐ x. This rule tells us that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. If you have an exponent inside a log, you can bring that exponent out front and multiply it by the log.

These properties are the keys to unlocking and simplifying logarithmic expressions. They allow us to manipulate logarithms in ways that make them easier to evaluate. By understanding and applying these rules, we can transform complex expressions into simpler ones, making them much easier to handle. So, keep these properties in mind as we move forward, and you'll see how powerful they can be in simplifying logarithms.

Applying Logarithmic Properties to the Expression

Okay, let's get our hands dirty and apply these logarithmic properties to the expression log₂ 20 + log₂ 5 - log₂ 4. Remember, our goal is to simplify this expression using the rules we just discussed. The first thing we notice is that we have a combination of addition and subtraction of logarithms with the same base (base 2). This is a perfect scenario for applying the product and quotient rules.

Let's start by tackling the addition part: log₂ 20 + log₂ 5. According to the product rule, the sum of two logarithms with the same base is equal to the logarithm of the product of their arguments. In other words, we can combine these two logarithms into a single logarithm by multiplying their arguments:

log₂ 20 + log₂ 5 = log₂ (20 * 5) = log₂ 100

Now our expression looks like this: log₂ 100 - log₂ 4. We've successfully simplified the addition part into a single logarithm. Next up, we have the subtraction of two logarithms with the same base. This is where the quotient rule comes into play. The quotient rule tells us that the difference of two logarithms with the same base is equal to the logarithm of the quotient of their arguments. So, we can combine these logarithms by dividing the first argument by the second:

log₂ 100 - log₂ 4 = log₂ (100 / 4) = log₂ 25

Awesome! We've further simplified our expression to log₂ 25. This is much cleaner and easier to work with than the original expression. We've successfully applied the product and quotient rules to combine the logarithms and reduce the expression to a single logarithm. But we're not quite done yet. The next step is to evaluate this simplified logarithm. We need to figure out what power we need to raise 2 to in order to get 25. This is where our understanding of logarithms comes full circle, and we can finally find the numerical value of our expression.

Evaluating the Simplified Logarithmic Expression

We've successfully simplified the original expression to log₂ 25. Now comes the exciting part: evaluating this logarithm. Remember, log₂ 25 asks the question: "To what power must we raise 2 to get 25?" This isn't as straightforward as some logarithms, like log₂ 8 (which is 3), because 25 isn't a perfect power of 2. So, we'll need to use a little bit of estimation and some tools to get a precise answer.

Let's start with some estimation. We know that 2⁴ = 16 and 2⁵ = 32. Since 25 falls between 16 and 32, we know that the value of log₂ 25 must be between 4 and 5. This gives us a good starting point. To get a more precise answer, we can use a calculator. Most calculators have a log function, but it's usually for base 10 logarithms (log₁₀), often written simply as "log." To evaluate a logarithm with a different base, like our base 2 logarithm, we need to use the change of base formula.

The change of base formula is a powerful tool that allows us to convert logarithms from one base to another. It states:

logₐ b = logₓ b / logₓ a

where a is the original base, b is the argument, and x is the new base we want to use. We can choose any base for x, but base 10 is often the most convenient because most calculators have a base 10 logarithm function. Applying the change of base formula to our expression, we get:

log₂ 25 = log₁₀ 25 / log₁₀ 2

Now, we can use a calculator to find the values of log₁₀ 25 and log₁₀ 2. Using a calculator, we find that:

log₁₀ 25 ≈ 1.3979

log₁₀ 2 ≈ 0.3010

Plugging these values back into our equation, we get:

log₂ 25 ≈ 1.3979 / 0.3010 ≈ 4.644

Therefore, log₂ 25 is approximately equal to 4.644. This aligns with our earlier estimation that the value should be between 4 and 5. We've now successfully evaluated the simplified logarithmic expression using the change of base formula and a calculator. Rounding to the Nearest Thousandths Place

The final step in our journey is to round our answer to the nearest thousandths place. We found that log₂ 25 is approximately 4.644. The thousandths place is the third digit after the decimal point. In this case, the digit in the thousandths place is 4. To round to the nearest thousandths place, we look at the digit to the right of the thousandths place, which is the ten-thousandths place. If that digit is 5 or greater, we round up the digit in the thousandths place. If it's less than 5, we leave the digit in the thousandths place as it is.

In our case, the value we calculated, 4.644, is already given to the thousandths place. If there were more digits after the 4, we would need to consider them for rounding. However, since we only have three digits after the decimal point, our answer is already rounded to the nearest thousandths place. Therefore, the final answer, rounded to the nearest thousandths place, is 4.644.

We've successfully navigated the entire process, from simplifying the original expression using logarithmic properties to evaluating the simplified expression using the change of base formula and a calculator, and finally rounding our answer to the nearest thousandths place. This demonstrates a comprehensive understanding of how to work with logarithms and solve logarithmic problems.

Conclusion

Alright, guys, we've reached the end of our logarithmic adventure! We started with the expression log₂ 20 + log₂ 5 - log₂ 4 and, step by step, we simplified it using the product and quotient rules of logarithms. Then, we evaluated the simplified expression log₂ 25 using the change of base formula and a calculator, arriving at an approximate value of 4.644. Finally, we rounded our answer to the nearest thousandths place, confirming our result as 4.644.

This process highlights the power and elegance of logarithmic properties. By understanding and applying these rules, we can transform complex expressions into manageable forms, making them much easier to evaluate. Logarithms are a fundamental concept in mathematics and have wide-ranging applications in various fields, including science, engineering, and finance. Mastering these skills will not only help you ace your math exams but also equip you with valuable tools for problem-solving in the real world.

Remember, practice makes perfect! The more you work with logarithms, the more comfortable and confident you'll become. So, keep exploring, keep practicing, and don't be afraid to tackle challenging problems. You've got this! If you ever get stuck, remember the steps we've covered today: simplify using logarithmic properties, evaluate using the change of base formula (if needed), and round to the required precision. And most importantly, have fun with it! Math can be a fascinating journey, and logarithms are just one of the many exciting destinations along the way.