Solve Logₓ125 = 3: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of logarithmic equations. Specifically, we're going to break down how to solve the equation ${\log_x 125 = 3}$. Don't worry if you find logarithms a bit tricky – we'll go through each step together. By the end of this guide, you'll not only know the answer but also understand why it's the answer. Let's get started!

Understanding Logarithms

Before we jump into solving our equation, let's quickly recap what logarithms are all about. Logarithms are essentially the inverse operation to exponentiation. Think of it this way: if you have an equation like ${b^y = x}$, the logarithm answers the question, "What exponent do I need to raise ${b}$ to, in order to get ${x}$?" This question is represented as ${\log_b x = y}$. In this notation:

• ${b}$ is the base of the logarithm.
• ${x}$ is the argument (the value we're taking the logarithm of).
• ${y}$ is the exponent (the logarithm itself).

To really nail this down, let’s consider a few examples. Imagine we have the expression ${2^3 = 8}$. In logarithmic form, this is written as ${\log_2 8 = 3}$. See how the base (2) remains the same, the exponent (3) becomes the result of the logarithm, and the result of the exponentiation (8) becomes the argument of the logarithm? Another example: ${10^2 = 100}$ translates to ${\log_{10} 100 = 2}$. This illustrates the core concept: a logarithm tells you the power to which you must raise the base to obtain the argument.

Now, why is understanding this fundamental relationship so important? Because it provides us with the key to unraveling logarithmic equations. When we're faced with an equation like ${\log_x 125 = 3}$, we're essentially being asked, "To what power must we raise ${x}$ to get 125 if the power is 3?" By converting the logarithmic equation back into its exponential form, we transform it into a more familiar algebraic problem, something we can solve using basic algebraic principles. So, keep this in mind: the ability to switch between logarithmic and exponential forms is your superpower when tackling these types of equations. With a solid grasp of this concept, you'll find that logarithms are not as intimidating as they might seem initially.

Converting Logarithmic to Exponential Form

Okay, now that we've refreshed our understanding of logarithms, let's get practical. The crucial step in solving the equation ${\log_x 125 = 3}$ is converting it from logarithmic form to exponential form. This transformation makes the equation much easier to handle. Remember the general relationship we discussed: ${\log_b x = y}$ is equivalent to ${b^y = x}$. This is the golden rule for converting between logarithmic and exponential expressions.

Let's apply this rule to our specific equation, ${\log_x 125 = 3}$. Here, we can identify the parts: the base ${b}$ is ${x}$, the argument ${x}$ (in the general form) is 125, and the logarithm ${y}$ is 3. Plugging these values into the exponential form ${b^y = x}$, we get:

${x^3 = 125}$

See how much simpler this looks? We've successfully transformed a logarithmic equation into a basic algebraic one. This conversion is the key because it allows us to use familiar algebraic techniques to isolate our variable, which in this case is ${x}$. The logarithmic form hides the variable within the logarithm, making it less accessible. The exponential form, however, brings the variable into the open, allowing us to directly manipulate it.

To further illustrate the power of this conversion, let's consider another example. Suppose we have ${\log_2 16 = 4}$. Converting this to exponential form using our rule, we get ${2^4 = 16}$, which is a straightforward exponential statement. This simple example highlights why the conversion is so effective: it transforms a logarithm, which can seem abstract, into a concrete exponential relationship that is often easier to understand and solve. So, whenever you encounter a logarithmic equation, your first instinct should be to convert it to exponential form. This one step can make the problem significantly more manageable.

Solving the Exponential Equation

Great! We've successfully transformed our logarithmic equation ${\log_x 125 = 3}$ into its exponential form, which is ${x^3 = 125}$. Now comes the fun part: actually solving for ${x}$. Our goal here is to isolate ${x}$ and find its value.

The equation ${x^3 = 125}$ is asking us, "What number, when multiplied by itself three times, equals 125?" One way to solve this is by recognizing that this is a cube root problem. We need to find the cube root of 125. In mathematical notation, this is represented as ${\sqrt[3]{125}}$.

If you're familiar with perfect cubes, you might already know that ${5^3 = 5 \times 5 \times 5 = 125}$. Therefore, the cube root of 125 is 5. If you don't immediately recognize this, don't worry! There are a couple of ways to approach finding the cube root. One method is to try factoring 125. We can see that 125 is divisible by 5, and ${125 = 5 \times 25}$. Then, we can further factor 25 as ${5 \times 5}$. So, we have ${125 = 5 \times 5 \times 5 = 5^3}$. This factorization clearly shows that the cube root of 125 is 5.

Another way to think about this is to systematically test numbers. Start with small integers and cube them until you find one that equals 125. For example:

• ${1^3 = 1}$
• ${2^3 = 8}$
• ${3^3 = 27}$
• ${4^3 = 64}$
• ${5^3 = 125}$

Bingo! We found that ${5^3 = 125}$, which confirms that the cube root of 125 is indeed 5. Therefore, the solution to our equation ${x^3 = 125}$ is ${x = 5}$.

In summary, to solve the exponential equation, we identified the need to find the cube root of 125. By either recognizing the perfect cube or using factorization, we determined that ${x = 5}$. This step is crucial because it gives us the numerical value of ${x}$ that satisfies the original logarithmic equation. Now, we're just one step away from completing our problem.

Checking the Solution

We've arrived at a potential solution: ${x = 5}$. But before we confidently declare victory, it's essential to check our answer. This is a crucial step in solving any equation, but especially so with logarithmic equations. Why? Because logarithms have certain restrictions on their arguments and bases, and we need to ensure our solution doesn't violate these rules.

The most important rule to remember is that the base of a logarithm must be positive and not equal to 1. The argument of a logarithm must also be positive. These restrictions come from the very definition of logarithms and their relationship to exponential functions. If we were to allow a negative base or a base of 1, we'd run into mathematical inconsistencies and undefined results.

So, let's plug our solution, ${x = 5}$, back into the original logarithmic equation, ${\log_x 125 = 3}$, to see if it holds true. Substituting ${x = 5}$, we get:

${\log_5 125 = 3}$

Now, we need to verify if this statement is correct. We can do this by converting it back to exponential form. Remember, ${\log_b x = y}$ is equivalent to ${b^y = x}$. So, ${\log_5 125 = 3}$ should be equivalent to ${5^3 = 125}$.

We already know that ${5^3 = 5 \times 5 \times 5 = 125}$. This confirms that our solution, ${x = 5}$, satisfies the equation. Moreover, our base, ${x = 5}$, is positive and not equal to 1, so it meets the necessary conditions for a valid logarithm.

By performing this check, we've not only confirmed the correctness of our solution but also reinforced our understanding of the relationship between logarithmic and exponential forms. This step is not just a formality; it's a vital part of the problem-solving process that helps prevent errors and builds confidence in our answer. So, always remember to check your solutions when dealing with logarithmic equations!

Final Answer

Phew! We've journeyed through the world of logarithms, converted equations, and solved for our unknown. After all that, it's time to state our final answer. We started with the equation ${\log_x 125 = 3}$, converted it to exponential form ${x^3 = 125}$, found the cube root of 125, and verified our solution. So, what is the value of ${x}$?

The solution to the equation ${\log_x 125 = 3}$ is:

${x = 5}$

Therefore, the correct answer from the options provided is C. 5.

We not only found the answer but also understood the process behind it. We learned how to convert logarithmic equations to exponential form, how to solve basic exponential equations, and the importance of checking our solutions. Understanding these steps will empower you to tackle a wide range of logarithmic problems. Keep practicing, and you'll become a logarithm pro in no time! Remember, math is a journey, not a destination. Enjoy the process of learning and problem-solving!