Solving 8 * 5^x = 48 A Detailed Explanation

Hey guys! Today, we're diving deep into an exciting mathematical problem: solving the exponential equation 8 \cdot 5^x = 48. This might seem a bit daunting at first, but don't worry, we'll break it down step by step. Exponential equations pop up frequently in various fields like finance, physics, and computer science, so mastering them is super important. Think compound interest, radioactive decay, or even the spread of information online – all these involve exponential growth or decay. So, buckle up, and let's get started on unraveling this equation!

1. Understanding Exponential Equations

Before we jump into solving exponential equations, let's make sure we're all on the same page about what they are. An exponential equation is essentially an equation where the variable appears in the exponent. Our equation, 8 \cdot 5^x = 48, is a classic example. The key here is that 'x' is the exponent of the base 5. Now, why are these equations so important? Well, exponential functions model phenomena where the rate of change is proportional to the current value. Imagine a population of bacteria doubling every hour – that's exponential growth in action! Understanding how to manipulate and solve these equations gives us powerful tools for predicting and analyzing these types of real-world scenarios.

To get a better handle on exponential equations, it's helpful to contrast them with other types of equations, like linear or quadratic equations. In a linear equation (e.g., 2x + 3 = 7), the variable appears with a power of 1. In a quadratic equation (e.g., x^2 + 4x + 4 = 0), the variable has a power of 2. Exponential equations, with the variable in the exponent, behave quite differently. This means we need a different set of techniques to solve them.

2. Isolating the Exponential Term

The first step in solving our equation, 8 \cdot 5^x = 48, is to isolate the exponential term, which in this case is 5^x. This means we need to get 5^x by itself on one side of the equation. To do this, we'll use basic algebraic manipulation. The golden rule of equation solving is that whatever you do to one side, you must do to the other. In our case, we can divide both sides of the equation by 8. This gives us: (8 \cdot 5^x) / 8 = 48 / 8, which simplifies to 5^x = 6. Now we're in business! We've successfully isolated the exponential term.

Isolating the exponential term is crucial because it sets the stage for using logarithms, which are the key to unlocking the value of 'x'. Think of logarithms as the inverse operation of exponentiation. Just like subtraction undoes addition, and division undoes multiplication, logarithms undo exponentiation. By isolating the exponential term, we're preparing to apply a logarithm that will effectively bring the exponent 'x' down to ground level, so to speak.

3. Applying Logarithms

Okay, we've got 5^x = 6. Now for the fun part: applying logarithms! As we mentioned earlier, logarithms are the inverse of exponentiation. The logarithm of a number to a given base is the exponent to which we must raise the base to produce that number. There are two main types of logarithms we often use: the common logarithm (base 10), denoted as log, and the natural logarithm (base e), denoted as ln. For our equation, we can use either one, but let's go with the natural logarithm (ln) for this example. The beauty of logarithms is that they allow us to bring the exponent down as a coefficient. This is thanks to a fundamental property of logarithms: log_b(a^c) = c \cdot log_b(a). In plain English, the logarithm of a number raised to a power is equal to the power times the logarithm of the number.

Applying the natural logarithm to both sides of our equation 5^x = 6, we get ln(5^x) = ln(6). Now, using the logarithmic property we just discussed, we can rewrite the left side as x \cdot ln(5) = ln(6). See what happened? The 'x' is no longer an exponent; it's a coefficient! We're one step closer to solving for 'x'. This step is the heart of solving exponential equations – it's where we transform the equation into a form we can easily solve using basic algebra.

4. Solving for x

We've reached the final stretch! We have the equation x \cdot ln(5) = ln(6). Our goal is to isolate 'x', so we need to get rid of the ln(5) that's multiplying it. How do we do that? You guessed it – division! We'll divide both sides of the equation by ln(5). This gives us (x \cdot ln(5)) / ln(5) = ln(6) / ln(5), which simplifies to x = ln(6) / ln(5). We've done it! We've solved for 'x'. Now, to get a numerical approximation, we can use a calculator to find the values of ln(6) and ln(5). ln(6) is approximately 1.7918, and ln(5) is approximately 1.6094. So, x ≈ 1.7918 / 1.6094 ≈ 1.1133.

Therefore, the solution to the equation 8 \cdot 5^x = 48 is approximately x ≈ 1.1133. To verify our solution, we can plug this value back into the original equation and see if it holds true. 8 \cdot 5^(1.1133) ≈ 8 \cdot 6.0001 ≈ 48.0008, which is very close to 48. The slight discrepancy is due to rounding errors. This final step of verifying our solution is always a good practice to ensure we haven't made any mistakes along the way.

5. Alternative Methods and Considerations

While we used natural logarithms to solve our equation, it's worth noting that we could have used common logarithms (base 10) as well. The process would be exactly the same, just with 'log' instead of 'ln'. The choice between natural and common logarithms is often a matter of preference or the specific context of the problem. Both will lead to the same answer. Another important point is that not all exponential equations have solutions that can be expressed in a simple, closed form. In some cases, the solution might be an irrational number that can only be approximated numerically. Also, be mindful of the domain of the exponential function. The base of the exponent must be positive, and if the base is 1, the equation might have infinitely many solutions or no solutions at all.

In summary, solving exponential equations involves isolating the exponential term, applying logarithms to both sides, and then solving for the variable. It's a powerful technique with applications in various fields. Practice is key to mastering this skill, so try tackling different exponential equations to build your confidence and understanding.