Logarithmic Expression Conversion: Find The Original!

Hey guys! Today, we're diving deep into the world of logarithms, specifically focusing on the change of base formula and how it transforms logarithmic expressions. We've got a fun problem where Tyler used this formula, and we need to figure out what the original expression looked like. So, buckle up, and let's get started!

Understanding the Change of Base Formula

Before we jump into the problem, let's quickly recap the change of base formula. This formula is super handy because it allows us to evaluate logarithms with any base using a calculator, which usually only has buttons for common logarithms (base 10) or natural logarithms (base e). The formula states:

$$\log_a b = \frac{\log_c b}{\log_c a}$$

Where:

• a is the original base of the logarithm.
• b is the argument (the number you're taking the logarithm of).
• c is the new base you're changing to (usually 10 or e).

In simpler terms, if you have a logarithm with a base you don't like, you can rewrite it as a fraction of two logarithms with a base you do like. The argument of the original logarithm becomes the argument of the numerator, and the original base becomes the argument of the denominator. Keep in mind the logarithm properties.

The beauty of this formula lies in its flexibility. You can choose any base c that's convenient for your calculations. Most often, we use base 10 (denoted as log) or base e (denoted as ln) because these are readily available on calculators. Essentially, the change of base formula provides a bridge between different logarithmic scales, enabling us to compute logarithms that would otherwise be difficult to evaluate directly. This is particularly useful when dealing with logarithms that have uncommon or inconvenient bases. Understanding and applying this formula correctly is a fundamental skill in logarithmic manipulations and problem-solving. It's a tool that simplifies complex calculations and allows for a deeper understanding of the relationships between different logarithmic expressions. So, make sure you've got this formula down pat – it's going to come in handy!

Analyzing Tyler's Transformation

Tyler applied the change of base formula and ended up with this expression:

$$\frac{\log \frac{1}{4}}{\log 12}$$

Our mission, should we choose to accept it, is to figure out what Tyler's original expression was before he used the formula. Let's compare this to the change of base formula we just discussed:

$$\log_a b = \frac{\log_c b}{\log_c a}$$

Notice that in Tyler's final expression:

• The numerator is $\log \frac{1}{4}$, which corresponds to $\log_c b$ in the formula.
• The denominator is $\log 12$, which corresponds to $\log_c a$ in the formula.

This means that $\frac{1}{4}$ played the role of b (the argument of the original logarithm), and 12 played the role of a (the original base). Therefore, Tyler's original expression must have been in the form:

$$\log_a b = \log_{12} \frac{1}{4}$$

Let's think about what this logarithm represents. It's asking the question: "To what power must we raise 12 to get $\frac{1}{4}$?" This might not be immediately obvious, but that's perfectly fine. The important thing is that we've correctly identified the original expression based on the change of base formula. By carefully matching the parts of the transformed expression to the components of the change of base formula, we were able to reverse the process and pinpoint the logarithm that Tyler started with. This highlights the power of understanding and applying mathematical formulas – they allow us to move between different representations of the same concept and solve problems in creative ways. So, give yourself a pat on the back for following along – you're one step closer to mastering logarithms!

Identifying the Correct Option

Now, let's look at the options provided and see which one matches our deduced original expression:

A. $\log_{\frac{1}{4}} 12$ B. $\log_{12} \frac{1}{4}$

Comparing these to our deduced original expression, $\log_{12} \frac{1}{4}$, we can clearly see that option B is the correct answer.

Option A has the base and argument switched. Remember, the base is the small number written below the "log," and the argument is the number you're taking the logarithm of. Switching them completely changes the meaning of the expression.

Therefore, the expression that Tyler could have started with is:

$$\log_{12} \frac{1}{4}$$

This exercise highlights the importance of careful observation and attention to detail when working with mathematical formulas. By meticulously comparing the given expression to the change of base formula, we were able to successfully identify the original logarithmic expression. This not only reinforces our understanding of the formula itself but also demonstrates the power of analytical thinking in problem-solving. So, always remember to take your time, double-check your work, and trust in your ability to apply the knowledge you've gained. With practice and perseverance, you'll become a master of logarithmic transformations and much more!

Key Takeaways

• The change of base formula is a powerful tool for evaluating logarithms with any base.
• Carefully compare the transformed expression to the formula to identify the original base and argument.
• Pay close attention to detail and avoid switching the base and argument of the logarithm.

So, there you have it, guys! We successfully unraveled Tyler's logarithmic transformation and found the original expression. Keep practicing with these types of problems, and you'll become a logarithm pro in no time! Remember always to practice and review the key concepts.