Logarithm Conversion: $\log _4 40$ Simplified

Hey everyone, and welcome back to Plastik Magazine! Today, we're diving deep into the fascinating world of logarithms, specifically tackling a problem that might look a little tricky at first glance: rewriting $\log _4 40$ as a quotient of two common logarithms in its simplest form. Don't worry, guys, it's not as complicated as it sounds, and by the end of this, you'll be a logarithm-changing pro! We'll break it down step-by-step, making sure you understand each part of the process. So, grab your notebooks, and let's get started on unraveling this mathematical puzzle.

Understanding Common Logarithms and the Change of Base Formula

Before we jump into solving $\log _4 40$, let's quickly chat about what common logarithms are and why they're so useful. A common logarithm is simply a logarithm with a base of 10. We usually write it as just 'log' without the base explicitly shown (so, $\log x$ is the same as $\log _{10} x$). These are super handy because our entire number system is based on 10, making calculations with them a bit more intuitive. Now, the real magic happens when we introduce the change of base formula. This formula is an absolute game-changer, allowing us to convert a logarithm from one base to any other base we like. It's like having a universal translator for logarithms! The formula states that for any positive numbers $a$, $b$, and $x$, where $a \neq 1$ and $b \neq 1$, we have:

$$\log _a x = \frac{\log _b x}{\log _b a}$$

See that? We can rewrite any logarithm ($\log _a x$) as a fraction (a quotient) of two new logarithms with a different base ($b$). The beauty of this formula is that we can choose any convenient base for $b$. And guess what? When we want to express a logarithm in terms of common logarithms, we simply choose $b=10$. So, applying the change of base formula with $b=10$ to our general logarithm $\log _a x$, we get:

$$\log _a x = \frac{\log _{10} x}{\log _{10} a}$$

Or, more commonly written as:

$$\log _a x = \frac{\log x}{\log a}$$

This is exactly what the problem is asking us to do: rewrite our given logarithm, $\log _4 40$, as a quotient of two common logarithms. We'll be using this formula as our main tool to break down the problem and arrive at the simplest form. Remember, the key here is understanding that this formula gives us the flexibility to work with logarithms in bases that are more convenient for calculation or for fulfilling specific problem requirements, like converting to common logs.

Applying the Change of Base Formula to $\log _4 40$

Alright, now that we've got the change of base formula fresh in our minds, let's apply it directly to our specific problem: $\log _4 40$. We want to rewrite this as a quotient of two common logarithms. Using the formula we just discussed, $\log _a x = \frac{\log x}{\log a}$, we can identify our parts:

• The original base, $a$, is 4.
• The number we're taking the logarithm of, $x$, is 40.

Substituting these values into the formula, we get:

$$\log _4 40 = \frac{\log 40}{\log 4}$$

And just like that, we've successfully rewritten $\log _4 40$ as a quotient of two common logarithms! This is the direct application of the change of base formula when the target base is 10 (the common logarithm). The numerator is the common logarithm of the original argument (40), and the denominator is the common logarithm of the original base (4). It's that straightforward! This step is crucial because it converts a logarithm with a base that might not be readily available on a standard calculator into an expression involving base-10 logarithms, which are standard. So, in essence, we've taken a logarithm that requires special handling and transformed it into a form that's much easier to work with or approximate using tools that are widely accessible. The goal isn't always to find a single numerical value at this stage, but to express it in a universally understood format, and that's precisely what the change of base formula facilitates for common logarithms. Keep this result handy, because the next step is to simplify it even further!

Simplifying the Logarithmic Expression

We've successfully converted $\log _4 40$ into $\frac{\log 40}{\log 4}$. Now, the problem asks us to write the answer in its simplest form. This means we need to see if we can simplify the numbers inside the logarithms. Let's look at the numbers 40 and 4. We know that 40 can be expressed in terms of factors, and importantly, it has a relationship with powers of 2, just like 4 does. Specifically, we can write 40 as $8 \times 5$. And we know that 8 is $2^3$. So, $40 = 2^3 \times 5$. For the denominator, we know that 4 is simply $2^2$.

Using the product rule of logarithms, which states that $\log (MN) = \log M + \log N$, we can rewrite the numerator, $\log 40$:

$$\log 40 = \log (8 \times 5) = \log 8 + \log 5$$

Since $8 = 2^3$, we can use the power rule of logarithms, which states that $\log (M^p) = p \log M$, to simplify $\log 8$:

$$\log 8 = \log (2^3) = 3 \log 2$$

So, our numerator becomes:

$$\log 40 = 3 \log 2 + \log 5$$

Now let's look at the denominator, $\log 4$. We know $4 = 2^2$. Using the power rule again:

$$\log 4 = \log (2^2) = 2 \log 2$$

Putting it all back together in our quotient form:

$$\frac{\log 40}{\log 4} = \frac{3 \log 2 + \log 5}{2 \log 2}$$

This is a much simpler form because we've broken down the original numbers into their prime factors and applied logarithmic properties. We've managed to express the entire logarithm in terms of logarithms of prime numbers (2 and 5) and constants. This is often what's meant by