Log Expression Value: Step-by-Step Calculation

by Andrew McMorgan 47 views

Hey Plastik Magazine readers! Today, let's dive into a fun mathematical problem involving logarithms. We're going to break down how to calculate the numerical value of a logarithmic expression. This might sound intimidating, but trust me, it's totally manageable once you understand the basic rules. So, let's get started and make logs less of a mystery, alright guys?

Understanding the Problem

First, let's clearly define the problem we're tackling. We need to find the value of the logarithmic expression: log(√(a * b^3) / c^9). But there's a catch! We're not given the actual values of a, b, and c directly. Instead, we're provided with their logarithms:

  • log a = 4
  • log b = -2
  • log c = 1

This means we'll need to use the properties of logarithms to manipulate the expression and substitute these values effectively. Think of it like a puzzle where we have to fit the pieces together. Our main goal here is to simplify the complex expression into smaller, manageable parts. This involves understanding key logarithmic properties and applying them step-by-step. Remember, logarithms are just exponents in disguise, so understanding exponent rules will also come in handy. We're essentially reversing the process of exponentiation to find the power to which a base must be raised to produce a given number. So, with our given values, we'll be working to find the value of the whole expression, not just individual variables.

Key Logarithmic Properties

Before we jump into the calculation, let's quickly review some essential logarithmic properties that we'll be using. These are the fundamental rules that govern how logarithms behave, and they're crucial for simplifying complex expressions like the one we're dealing with. Understanding these properties is like having the right tools in your toolbox – they allow you to tackle any logarithmic problem with confidence. So, let's make sure we're all on the same page and have these properties down!

  1. Product Rule: log(xy) = log x + log y. This property tells us that the logarithm of a product is equal to the sum of the logarithms of the individual factors. For example, log(2*3) is the same as log(2) + log(3). This rule is super useful for breaking down complex expressions into simpler ones.
  2. Quotient Rule: log(x/y) = log x - log y. This rule is similar to the product rule but applies to division. The logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator. So, log(10/2) is the same as log(10) - log(2). This is handy for dealing with fractions inside logarithms.
  3. Power Rule: log(x^n) = n log x. This is a big one! The logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. This means log(2^3) is the same as 3 * log(2). This rule is incredibly helpful for dealing with exponents inside logarithms and is one we'll use extensively in this problem.
  4. Root Rule: log(√x) = (1/2) log x. This is a special case of the power rule. A square root is the same as raising something to the power of 1/2. So, the logarithm of the square root of x is equal to one-half times the logarithm of x. This is exactly what we need to deal with the square root in our original problem!

Step-by-Step Calculation

Okay, now that we've refreshed our memory on the key logarithmic properties, let's put them into action and solve the problem step-by-step. This is where we'll see how these rules make simplifying the expression much easier. We'll take it one step at a time, so you can follow along and understand each transformation. Remember, the key is to break down the complex expression into smaller, manageable parts using the properties we just discussed. So, grab your calculators (just in case) and let's get to work!

  1. Apply the Root Rule:

The first thing we need to do is tackle the square root in the expression. Remember the root rule? log(√x) = (1/2) log x. So, we can rewrite our expression as:

(1/2) log (a * b^3) / c^9

This step helps us remove the square root, making the expression easier to manipulate using other properties.

  1. Apply the Quotient Rule:

Now we have a fraction inside the logarithm. Time to use the quotient rule: log(x/y) = log x - log y. Applying this to our expression, we get:

(1/2) [log (a * b^3) - log c^9]

Notice that the (1/2) is still outside the brackets, as it applies to the entire expression. This step separates the numerator and denominator, making it clearer to deal with each part individually.

  1. Apply the Product Rule:

Inside the first logarithm, we have a product: a * b^3. Let's use the product rule: log(xy) = log x + log y. This gives us:

(1/2) [log a + log b^3 - log c^9]

Now we've broken down the product into individual logarithms, further simplifying the expression.

  1. Apply the Power Rule:

We have exponents in our logarithms (b^3 and c^9). Time for the power rule: log(x^n) = n log x. Applying this, we get:

(1/2) [log a + 3 log b - 9 log c]

This step is crucial as it allows us to bring the exponents down as coefficients, making it easy to substitute the given values.

  1. Substitute the Given Values:

Now comes the fun part! We know log a = 4, log b = -2, and log c = 1. Let's substitute these values into our expression:

(1/2) [4 + 3(-2) - 9(1)]

We've now replaced the logarithmic terms with their numerical values, bringing us closer to the final answer.

  1. Simplify the Expression:

Now it's just basic arithmetic. Let's simplify the expression inside the brackets:

(1/2) [4 - 6 - 9] (1/2) [-11]

  1. Calculate the Final Result:

Finally, multiply by (1/2):

-11/2 = -5.5

So, the numerical value of the logarithmic expression log(√(a * b^3) / c^9) is -5.5. Ta-da! We've successfully navigated through the logarithmic maze.

Conclusion

There you have it! We've successfully calculated the numerical value of the logarithmic expression by breaking it down step-by-step using key logarithmic properties. Remember, the trick is to identify the relevant rules and apply them systematically. Logarithms might seem tricky at first, but with practice, you'll become a pro in no time. So, keep practicing, keep exploring, and don't be afraid to tackle those challenging mathematical problems. Until next time, guys! Keep it stylish and stay curious!