Single Logarithm: Simplify Log Expressions Easily!

Hey guys! Today, we're going to tackle a common problem in mathematics: expressing a combination of logarithmic expressions as a single logarithm. This is a super useful skill when you're simplifying equations, solving for variables, or just trying to make things look cleaner. We'll break down the steps using the properties of logarithms and work through an example together. So, let's dive in and get those logarithmic expressions simplified!

Before we jump into the example, it's really important to understand the properties of logarithms. These properties are the tools we'll use to combine and simplify logarithmic expressions. Let's review the three key properties we'll be using:

1. The Power Rule: This rule states that $\log_b(x^p) = p \log_b(x)$. Basically, if you have a logarithm with an exponent inside, you can move the exponent to the front as a coefficient, and vice versa. This is super handy for dealing with coefficients in front of logarithms. For example, if we have $3 \log_6(x+8)$, the power rule allows us to rewrite this as $\log_6((x+8)^3)$. We're essentially taking the coefficient 3 and making it the exponent of the expression inside the logarithm. This is a fundamental step in combining logarithms because it allows us to get rid of coefficients that are standing in the way of applying other properties.

2. The Product Rule: The product rule tells us that $\log_b(mn) = \log_b(m) + \log_b(n)$. In plain English, the logarithm of a product is equal to the sum of the logarithms of the individual factors. This property is crucial for combining logarithms that are added together. Imagine you have two logarithms with the same base that are being added; you can combine them into a single logarithm by multiplying their insides. For instance, if we had $\log_2(5) + \log_2(3)$, we could use the product rule to combine this into $\\log_2(5 \times 3) = \log_2(15)$. This simplifies the expression and makes it easier to work with. Remember, the base b must be the same for this rule to apply!

3. The Quotient Rule: This rule is similar to the product rule but deals with division. It states that $\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)$. So, the logarithm of a quotient is equal to the difference of the logarithms. This rule is perfect for combining logarithms that are being subtracted. If you see two logarithms with the same base being subtracted, you can combine them into a single logarithm by dividing the inside of the first logarithm by the inside of the second logarithm. For example, if we see $\log_5(10) - \log_5(2)$, we can use the quotient rule to combine it into $\log_5(\frac{10}{2}) = \log_5(5)$. This is another powerful tool for simplifying expressions and getting them into a more manageable form. Like the product rule, the base b must be the same for this rule to work.

Let's put these properties into action with an example. Our goal is to write the following expression as a single logarithm:

$$3 \log_6(x+8) - \log_6(x-12) - \log_6(x-6)$$

It might look a little intimidating at first, but don't worry! We'll take it step by step. The key is to apply the logarithmic properties in the correct order. We'll start by using the power rule to deal with the coefficient, then we'll use the quotient rule to handle the subtraction.

Step 1: Apply the Power Rule

The first thing we need to do is get rid of that coefficient 3 in front of the first logarithm. Remember the power rule? It states that $\log_b(x^p) = p \log_b(x)$. We can use this rule in reverse to move the coefficient 3 inside the logarithm as an exponent.

So, we rewrite $3 \log_6(x+8)$ as $\log_6((x+8)^3)$. Now, our expression looks like this:

$$\\log_6((x+8)^3) - \log_6(x-12) - \log_6(x-6)$$

Great! We've taken care of the coefficient. Now we can move on to the next step, which involves dealing with the subtraction using the quotient rule.

Step 2: Apply the Quotient Rule

Now we have a series of logarithms being subtracted. Remember the quotient rule? It tells us that $\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)$. We can use this rule to combine the logarithms. Since we have two subtractions, we'll apply the quotient rule twice. First, let's combine the first two terms:

$$\\log_6((x+8)^3) - \log_6(x-12)$$

Using the quotient rule, we can rewrite this as:

$$\\log_6(\frac{(x+8)^3}{x-12})$$

Now our expression looks like this:

$$\\log_6(\frac{(x+8)^3}{x-12}) - \log_6(x-6)$$

We have one more subtraction to take care of. Let's apply the quotient rule again to combine the remaining terms:

$$\\log_6(\frac{(x+8)^3}{x-12}) - \log_6(x-6)$$

Applying the quotient rule, we get:

$$\\log_6(\frac{\frac{(x+8)^3}{x-12}}{x-6})$$

This looks a bit messy, but we can simplify the fraction inside the logarithm. To divide by $(x-6)$, we can multiply by its reciprocal, which is $\frac{1}{x-6}$. So, we rewrite the expression as:

$$\\log_6(\frac{(x+8)^3}{x-12} \times \frac{1}{x-6})$$

Now, we can combine the fractions by multiplying the numerators and the denominators:

$$\\log_6(\frac{(x+8)^3}{(x-12)(x-6)})$$

Step 3: Simplify (If Possible)

At this point, we have expressed the original expression as a single logarithm. The expression inside the logarithm is a fraction with a cubic term in the numerator and a product of two linear terms in the denominator.

$$\\log_6(\frac{(x+8)^3}{(x-12)(x-6)})$$

In some cases, you might be able to simplify this further by expanding the numerator and denominator and looking for common factors. However, in this particular example, there aren't any obvious simplifications we can make. So, we can leave the expression as it is.

So, after applying the power rule and the quotient rule, we've successfully expressed the original expression as a single logarithm:

$$3 \log_6(x+8) - \log_6(x-12) - \log_6(x-6) = \log_6(\frac{(x+8)^3}{(x-12)(x-6)})$$

This is our final answer! We've taken a complex combination of logarithmic expressions and simplified it into a single, more manageable logarithm.

Expressing logarithmic expressions as a single logarithm is a valuable skill in mathematics. By understanding and applying the power, product, and quotient rules, you can simplify complex expressions and make them easier to work with. Remember to take it step by step, apply the rules in the correct order, and don't be afraid to simplify along the way. With a little practice, you'll be a pro at combining logarithms in no time! Keep up the great work, guys! You've got this!