Simplifying Logarithmic Expressions ⁵log 150 - ⁵log 24 + ⁵log 4

Hey guys! Ever stumbled upon a logarithmic expression that looks like a puzzle? Well, today we're going to break down one of those puzzles together. We're diving into simplifying logarithmic expressions, and our specific problem is ⁵log 150 - ⁵log 24 + ⁵log 4. Don't worry, it sounds more intimidating than it actually is! We'll take it step-by-step, making sure you understand each move we make. So, grab your thinking caps, and let's get started on this logarithmic adventure!

Understanding Logarithms: The Basics

Before we jump into solving the expression, let's quickly recap what logarithms are all about. At its core, a logarithm is simply the inverse operation of exponentiation. Think of it this way: if 2³ = 8, then log₂ 8 = 3. The logarithm (log) tells you what exponent you need to raise the base (in this case, 2) to, in order to get a certain number (in this case, 8). So, the logarithm base 2 of 8 is 3 because 2 raised to the power of 3 equals 8.

The base of the logarithm is the small number written below the "log." In our problem, the base is 5 (⁵log). The number we're taking the logarithm of is called the argument. For instance, in ⁵log 150, 150 is the argument. Now, why is this important? Because understanding the relationship between bases, exponents, and arguments is crucial for manipulating logarithmic expressions. We need to be fluent in converting between exponential and logarithmic forms. For example, if we have bˣ = y, the logarithmic form is log_b y = x. This conversion is the key to unlocking many logarithmic puzzles. Another vital concept to grasp is the idea of common logarithms (base 10) and natural logarithms (base e). While our problem uses base 5, knowing about these common bases helps in understanding the broader landscape of logarithms. Remember, logarithms are not just abstract math concepts; they have real-world applications in fields like science, engineering, and finance. Understanding the basics thoroughly will not only help you solve expressions like this but also give you a strong foundation for more advanced mathematical concepts. So, let’s keep these basics in mind as we move forward and tackle our simplification challenge.

Key Logarithmic Properties: The Tools We'll Use

Okay, now that we've refreshed our understanding of the basics, let's equip ourselves with the essential tools for simplifying logarithmic expressions: the logarithmic properties. These properties are like secret codes that allow us to rewrite and manipulate logarithmic expressions into simpler forms. There are three main properties we'll be using today, so let’s get familiar with them.

First up is the product rule: log_b (x y) = log_b x + log_b y. This rule tells us 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 numbers multiplied together, you can split it into two separate logs added together. For example, ⁵log (2 * 75) can be rewritten as ⁵log 2 + ⁵log 75. This property is super handy when you want to break down a complex argument into simpler parts.

Next, we have the quotient rule: log_b (x / y) = log_b x - log_b y. This rule is the counterpart to the product rule, dealing with division instead of multiplication. It states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. So, if you see a logarithm of a fraction, you can separate it into two logs subtracted from each other. For instance, ⁵log (150 / 24) can be expressed as ⁵log 150 - ⁵log 24. This rule is particularly useful when you have fractions inside your logarithmic expressions.

Lastly, we have the power rule: log_b (x^p) = p log_b x. This rule lets us deal with exponents inside logarithms. It says that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. So, if you have an exponent inside a log, you can bring it down as a coefficient. For example, ⁵log (4²) can be rewritten as 2 * ⁵log 4. This rule is incredibly helpful when you need to simplify expressions involving exponents.

These three properties – product, quotient, and power – are the cornerstones of simplifying logarithmic expressions. Mastering them is crucial for solving problems like the one we're tackling today. Remember, practice makes perfect, so the more you use these properties, the more comfortable you'll become with them. Keep these tools in your mental toolkit, and let's see how we can apply them to our specific problem.

Applying the Properties: Step-by-Step Solution

Alright, with our logarithmic tools sharpened and ready, let's dive into simplifying the expression: ⁵log 150 - ⁵log 24 + ⁵log 4. Remember, the key here is to apply the properties we just discussed in a strategic way. So, let's break it down step by step.

Step 1: Combine the logarithms using the quotient and product rules

The first thing we notice is that we have a combination of subtraction and addition of logarithms with the same base (base 5). This is a perfect scenario for applying the quotient and product rules. Remember, subtraction of logarithms can be combined into a single logarithm using the quotient rule, and addition can be combined using the product rule. So, let’s tackle the subtraction first. We have ⁵log 150 - ⁵log 24. According to the quotient rule, this can be rewritten as ⁵log (150 / 24). Now, we have a single logarithm: ⁵log (150 / 24) + ⁵log 4. Next, we see the addition of logarithms. We can use the product rule to combine these into a single logarithm. The product rule tells us that ⁵log (150 / 24) + ⁵log 4 is the same as ⁵log [(150 / 24) * 4]. So, we've now combined our expression into a single logarithm: ⁵log [(150 / 24) * 4].

Step 2: Simplify the argument inside the logarithm

Now that we have a single logarithm, our next step is to simplify the argument inside the logarithm. We have ⁵log [(150 / 24) * 4]. Let's simplify the fraction and multiplication. First, let's simplify inside the brackets: (150 / 24) * 4. We can rewrite this as (150 * 4) / 24. Multiplying 150 by 4 gives us 600, so we have 600 / 24. Now, let's simplify this fraction. 600 divided by 24 is 25. So, our expression now looks like this: ⁵log 25. We've successfully simplified the argument inside the logarithm.

Step 3: Evaluate the logarithm

We're almost there! Our final step is to evaluate the simplified logarithm: ⁵log 25. Remember, a logarithm asks the question: