Solving Exponential Equations: Find X In 16 * 10^(5x) = 96

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Hey guys! Let's dive into solving an exponential equation today. We've got a fun one: 16 * 10^(5x) = 96. Our mission is to find the value of x, and we're going to round our final answer to the nearest hundredth. So, grab your calculators and let’s get started!

Understanding Exponential Equations

First off, what exactly is an exponential equation? Simply put, it's an equation where the variable appears in the exponent. In our case, that's the 5x sitting up there in the power of 10. Solving these equations involves a few key steps, and we'll break them down nice and easy.

The main goal when tackling an exponential equation is to isolate the exponential term. This means getting the part with the exponent (in our case, 10^(5x)) all by itself on one side of the equation. Once we've done that, we can use logarithms to bring the exponent down and solve for our variable.

Logarithms are basically the inverse operation of exponentiation. Think of it like this: if exponentiation is like raising a number to a power, logarithms are like figuring out what power you need to raise a number to in order to get a certain result. They might sound intimidating, but they're really just a tool to help us solve these types of problems. We'll use them to undo the exponent and get to that x.

The importance of understanding exponential equations extends far beyond just solving textbook problems. They pop up in all sorts of real-world situations, from calculating compound interest to modeling population growth and radioactive decay. Being able to manipulate and solve these equations is a valuable skill in many fields, including finance, science, and engineering.

So, let's keep this in mind as we work through our equation. We're not just plugging in numbers and churning out an answer; we're building a foundation for understanding a powerful mathematical concept that has wide-ranging applications. Let's get back to the problem at hand and see how we can put these ideas into practice.

Step-by-Step Solution

1. Isolate the Exponential Term

Our first job is to get that 10^(5x) term all by its lonesome on one side of the equation. Currently, it's being multiplied by 16. So, what’s the opposite of multiplying by 16? Dividing by 16! Let's do that to both sides of the equation:

16 * 10^(5x) = 96
(16 * 10^(5x)) / 16 = 96 / 16
10^(5x) = 6

Great! Now we have the exponential term isolated. The equation looks much simpler now, doesn't it? We're one step closer to finding x.

2. Apply Logarithms

Now comes the fun part where we use logarithms. Since we have a base of 10 in our exponential term (10^(5x)), the most convenient logarithm to use is the common logarithm, which is just the logarithm with base 10. It's often written as "log" without any subscript, and it's a standard function on most calculators. Applying the logarithm to both sides of the equation allows us to bring down the exponent:

log(10^(5x)) = log(6)

The cool thing about logarithms is that they have a property that lets us move exponents out in front as multipliers. This property is super useful for solving exponential equations. It says that log(a^b) = b * log(a). So, we can rewrite the left side of our equation as:

5x * log(10) = log(6)

But wait, there's more simplification we can do! Remember that log(10) (which is log base 10 of 10) is just equal to 1. This is because 10 raised to the power of 1 is 10. So, our equation simplifies even further:

5x * 1 = log(6)
5x = log(6)

We're getting closer and closer to solving for x. The logarithm has done its job of getting the exponent down, and now we just have a simple algebraic equation to deal with.

3. Solve for x

We're in the home stretch now! We have 5x = log(6). To get x by itself, we simply need to divide both sides of the equation by 5:

x = log(6) / 5

Now, we need to grab our calculators to find the value of log(6) and then divide it by 5. Make sure you're using the common logarithm (log base 10) function on your calculator. You should get something like:

x ≈ 0.77815 / 5

4. Round to the Nearest Hundredth

The problem asked us to round our answer to the nearest hundredth. That means we need to look at the digit in the thousandths place (the third digit after the decimal point) to decide whether to round up or down. In this case, we have:

x ≈ 0.77815

The digit in the thousandths place is 8, which is 5 or greater, so we round the hundredths digit up. Therefore, our final answer is:

x ≈ 0.16

Final Answer

So, the solution to the equation 16 * 10^(5x) = 96, rounded to the nearest hundredth, is x ≈ 0.16.

Practice Makes Perfect

Solving exponential equations might seem tricky at first, but with a little practice, you'll get the hang of it. Remember the key steps:

  1. Isolate the exponential term.
  2. Apply logarithms to both sides.
  3. Use logarithm properties to bring down the exponent.
  4. Solve for the variable.
  5. Round your answer if necessary.

The more you practice, the more comfortable you'll become with these steps. Try tackling some similar problems on your own. You can even change the numbers in this equation and solve it again to test your understanding. Keep practicing, and you'll be an exponential equation-solving pro in no time!

Understanding the nuances of logarithmic and exponential functions is paramount in various STEM fields. Whether it's determining the decay rate of a radioactive substance or calculating the growth of a bacterial colony, these equations serve as the backbone of numerous scientific calculations. Furthermore, in the realm of finance, understanding exponential growth is crucial for making informed decisions about investments and loans. The ability to accurately solve such equations not only enhances problem-solving skills but also provides a deeper insight into the mathematical underpinnings of the world around us.

So, keep up the great work, guys! Math is like a muscle – the more you use it, the stronger it gets. And who knows? Maybe next time we'll tackle an even more challenging exponential equation!

Real-World Applications

Exponential equations aren't just abstract math problems; they pop up in all sorts of real-world situations! Let's explore a few examples to see how these equations are used in practice.

1. Compound Interest

One of the most common applications of exponential equations is in calculating compound interest. When you deposit money into a savings account or take out a loan, the interest often compounds over time, meaning that the interest earned in one period is added to the principal, and then the next interest calculation is based on the new, higher amount. The formula for compound interest is:

A = P (1 + r/n)^(nt)

Where:

  • A = the future value of the investment/loan, including interest
  • P = the principal investment amount (the initial deposit or loan amount)
  • r = the annual interest rate (as a decimal)
  • n = the number of times that interest is compounded per year
  • t = the number of years the money is invested or borrowed for

This formula is a classic example of an exponential equation. If you want to figure out how long it will take for your investment to reach a certain value, you'll need to solve for t, which is in the exponent. This involves using logarithms, just like we did in our original problem.

2. Population Growth

Exponential equations are also used to model population growth. In many cases, populations tend to grow exponentially, meaning that the growth rate is proportional to the current population size. A simple model for population growth is:

P(t) = Pâ‚€ * e^(kt)

Where:

  • P(t) = the population at time t
  • Pâ‚€ = the initial population
  • e = the base of the natural logarithm (approximately 2.71828)
  • k = the growth rate constant
  • t = time

This equation is used to model everything from human populations to bacteria cultures. If you know the initial population and the growth rate, you can use this equation to predict the population at any future time. Conversely, if you know the population at two different times, you can solve for the growth rate constant k.

3. Radioactive Decay

Another important application of exponential equations is in modeling radioactive decay. Radioactive substances decay over time, and the amount of substance remaining decreases exponentially. The equation for radioactive decay is:

N(t) = N₀ * e^(-λt)

Where:

  • N(t) = the amount of substance remaining at time t
  • Nâ‚€ = the initial amount of substance
  • e = the base of the natural logarithm (approximately 2.71828)
  • λ = the decay constant
  • t = time

This equation is used in various fields, including nuclear physics, chemistry, and geology. For example, carbon-14 dating, a technique used to determine the age of ancient artifacts, relies on the exponential decay of carbon-14.

4. Viral Spread

In recent years, we've all become more aware of how exponential equations can be used to model the spread of viruses and diseases. In the early stages of an outbreak, the number of infected individuals can grow exponentially. Epidemiologists use these models to make predictions about the spread of a disease and to develop strategies for controlling it.

The ability to solve exponential equations is crucial for understanding and addressing these types of real-world problems. By mastering these mathematical tools, we can gain insights into a wide range of phenomena and make informed decisions in various aspects of our lives.

Conclusion

So, there you have it! We've successfully solved the exponential equation 16 * 10^(5x) = 96, and we've rounded our answer to the nearest hundredth. We've also explored some real-world applications of exponential equations, from compound interest to population growth and radioactive decay. Hopefully, this has given you a better appreciation for the power and versatility of these equations.

Remember, math is not just about memorizing formulas and procedures; it's about understanding the underlying concepts and applying them to solve problems. Keep practicing, keep exploring, and keep challenging yourselves. The world of mathematics is vast and fascinating, and there's always something new to learn! So, until next time, happy solving!