Solving Log Base 2 Of (3t+1) = 16 A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving deep into the world of logarithmic equations, specifically tackling the equation ${\log _2 (3t+1) = 16}$. If you've ever felt a bit puzzled by logs, don't worry – we're going to break it down step-by-step, making sure you not only understand the solution but also the why behind it. So, grab your calculators and let's get started!

Understanding Logarithms: The Basics

Before we jump into solving the equation, let's quickly recap what logarithms are all about. In simple terms, a logarithm answers the question: "To what power must we raise the base to get a certain number?" Mathematically, if we have ${\log_b a = c}$, this means that ${b^c = a}$. Here, ${b}$ is the base, ${a}$ is the argument, and ${c}$ is the exponent (or the logarithm itself).

To really grasp this, let's look at an example. Consider ${\log_2 8 = 3}$. This is asking, "To what power must we raise 2 to get 8?" The answer, of course, is 3, since ${2^3 = 8}$. Understanding this fundamental relationship between logarithms and exponents is crucial for solving logarithmic equations.

Now, let's talk about the key components in our equation: ${\log _2 (3t+1) = 16}$. Here, the base is 2, the argument is ${(3t+1)}$, and the logarithm is 16. Our mission is to find the value of ${t}$ that satisfies this equation. The first step in solving any logarithmic equation is to convert it into its exponential form. This makes the equation much easier to manipulate and solve.

Converting a logarithmic equation to exponential form involves using the definition we just discussed: ${\log_b a = c}$ is equivalent to ${b^c = a}$. Applying this to our equation, ${\log _2 (3t+1) = 16}$, we get:

${2^{16} = 3t + 1}$

See how we've transformed the logarithmic equation into a simple algebraic one? This is a significant step forward. Now, we have a clear path to isolate ${t}$ and find its value. Remember, the key is to understand the relationship between logarithms and exponents – they are two sides of the same coin.

Step-by-Step Solution of ${\log _2 (3t+1) = 16}$

Okay, guys, let's break down the solution to our equation ${\log _2 (3t+1) = 16}$ step by step. We've already made a fantastic start by converting the logarithmic equation into its exponential form:

${2^{16} = 3t + 1}$

This is where the real fun begins! Our next goal is to isolate ${t}$. To do this, we'll follow a series of algebraic manipulations that will lead us to the solution.

Step 1: Calculate ${2^{16}}$

First things first, let's figure out what ${2^{16}}$ actually is. If you've got a calculator handy, this is a piece of cake. ${2^{16}}$ means 2 multiplied by itself 16 times. Doing the math, we find that:

${2^{16} = 65536}$

So, our equation now looks like this:

${65536 = 3t + 1}$

We've simplified things quite a bit, haven't we? Now, we're one step closer to getting ${t}$ all by itself.

Step 2: Isolate the Term with ${t}$

To get the term with ${t}$ alone on one side of the equation, we need to get rid of that pesky +1. We can do this by subtracting 1 from both sides of the equation. This keeps the equation balanced and moves us closer to our goal.

${65536 - 1 = 3t + 1 - 1}$

${65535 = 3t}$

Awesome! We've successfully isolated the term with ${t}$. Now, all that's left is to get ${t}$ by itself.

Step 3: Solve for ${t}$

To finally solve for ${t}$, we need to undo the multiplication by 3. The opposite of multiplication is division, so we'll divide both sides of the equation by 3.

${\frac{65535}{3} = \frac{3t}{3}}$

${t = 21845}$

And there we have it! We've found the value of ${t}$ that satisfies the equation. The solution is ${t = 21845}$. It might seem like a big number, but that's perfectly okay. Logarithmic equations can sometimes have solutions that are quite large or small.

Step 4: Verification (Always a Good Idea!)

Before we celebrate too much, let's make sure our solution is correct. The best way to do this is to plug our value of ${t}$ back into the original equation and see if it holds true.

Original equation: ${\log _2 (3t+1) = 16}$

Substitute ${t = 21845}$:

${\log _2 (3(21845)+1) = 16}$

${\log _2 (65535+1) = 16}$

${\log _2 (65536) = 16}$

Now, we know that ${2^{16} = 65536}$, so:

${\log _2 (2^{16}) = 16}$

This is indeed true! Our solution checks out. We've successfully solved the logarithmic equation.

Common Mistakes to Avoid When Solving Logarithmic Equations

When you are solving logarithmic equations, it's easy to stumble if you're not careful. Let's shine a spotlight on some common pitfalls so you can steer clear of them. Knowing what mistakes to avoid is just as important as knowing the steps to solve the equation.

Mistake 1: Forgetting the Domain of Logarithms

This is a big one. Logarithms are only defined for positive arguments. In other words, you can't take the logarithm of a negative number or zero. So, before you even start solving, make sure that the argument of the logarithm is positive. In our equation, ${\log _2 (3t+1) = 16}$, the argument is ${(3t+1)}$. We need to ensure that:

${3t + 1 > 0}$

Solving this inequality gives us:

${3t > -1}$

${t > -\frac{1}{3}}$

So, any solution we find must be greater than ${-\frac{1}{3}}$. If we get a solution that doesn't satisfy this condition, we know it's an extraneous solution (a solution that doesn't actually work in the original equation). This is a critical step to remember.

Mistake 2: Incorrectly Converting to Exponential Form

As we discussed earlier, converting a logarithmic equation to exponential form is a key step. But it's also a place where mistakes can easily happen. Remember the relationship: ${\log_b a = c}$ is equivalent to ${b^c = a}$. Make sure you identify the base, argument, and logarithm correctly before you make the conversion. A slight mix-up here can throw off your entire solution.

Mistake 3: Algebraic Errors

Once you've converted the equation to exponential form, you're dealing with algebra. And algebraic errors are always a possibility. Be careful with your arithmetic, especially when dealing with exponents and negative signs. Double-check each step to make sure you haven't made any mistakes.

Mistake 4: Not Checking for Extraneous Solutions

We touched on this earlier, but it's worth repeating: always, always check your solutions. Plug your solution back into the original logarithmic equation to make sure it works. This is especially important when dealing with logarithmic equations, as extraneous solutions can pop up due to the domain restrictions.

Mistake 5: Misunderstanding Logarithmic Properties

Logarithms have some useful properties that can help simplify equations. But if you don't understand these properties, you might use them incorrectly. For example, remember that ${\log_b (xy) = \log_b x + \log_b y}$, not ${\log_b x \cdot \log_b y}$. A solid understanding of these properties is crucial for solving more complex logarithmic equations.

By being aware of these common mistakes, you'll be well-equipped to tackle logarithmic equations with confidence. Remember, practice makes perfect, so keep working at it!

Real-World Applications of Logarithms

Okay, so we've conquered the equation ${\log _2 (3t+1) = 16}$, but you might be wondering, "Where does this stuff actually get used in the real world?" Well, guys, logarithms are incredibly versatile and pop up in all sorts of unexpected places. Let's explore some fascinating applications.

1. Measuring Earthquakes: The Richter Scale

One of the most well-known applications of logarithms is in the Richter scale, which measures the magnitude of earthquakes. The Richter scale is a base-10 logarithmic scale, meaning that each whole number increase on the scale represents a tenfold increase in the amplitude of the seismic waves. For example, an earthquake with a magnitude of 6 is ten times stronger than an earthquake with a magnitude of 5. This logarithmic scale allows us to represent a wide range of earthquake intensities in a manageable way. This is a classic example of the power of logarithms in handling large numbers.

2. Sound Intensity: Decibels

Sound intensity is another area where logarithms play a crucial role. The loudness of sound is measured in decibels (dB), which is a logarithmic unit. The decibel scale is based on the logarithm of the ratio of the sound intensity to a reference intensity. Just like the Richter scale, this logarithmic scale allows us to represent a wide range of sound intensities, from the quietest whisper to the loudest rock concert. The human ear perceives sound logarithmically, which is why the decibel scale is such a natural fit. A deeper understanding of logarithms helps us appreciate how we perceive the world around us.

3. Chemistry: pH Scale

In chemistry, the pH scale is used to measure the acidity or alkalinity of a solution. The pH is defined as the negative logarithm (base 10) of the concentration of hydrogen ions (${H^+}$) in a solution. A pH of 7 is neutral, pH values less than 7 are acidic, and pH values greater than 7 are alkaline. The logarithmic nature of the pH scale means that a small change in pH represents a significant change in acidity or alkalinity. For instance, a solution with a pH of 3 is ten times more acidic than a solution with a pH of 4.

4. Computer Science: Algorithm Analysis

Logarithms are also fundamental in computer science, particularly in the analysis of algorithms. Many efficient algorithms have a time complexity that is logarithmic, meaning that the time it takes to run the algorithm increases logarithmically with the size of the input. For example, the binary search algorithm, which is used to find a specific element in a sorted list, has a logarithmic time complexity. This means that the algorithm can search through a vast amount of data very quickly. Understanding logarithms is essential for designing and analyzing efficient algorithms.

5. Finance: Compound Interest

In the world of finance, logarithms are used in calculations involving compound interest. Compound interest is the interest earned on both the principal amount and the accumulated interest. The formula for compound interest involves logarithms, which can be used to determine the time it takes for an investment to reach a certain value. This is a powerful tool for financial planning and investment analysis.

These are just a few examples of the many real-world applications of logarithms. From measuring earthquakes to analyzing algorithms, logarithms are an indispensable tool in science, engineering, and finance. So, the next time you encounter a logarithmic scale or calculation, you'll have a better appreciation for the underlying math.

Conclusion: Mastering Logarithmic Equations

Alright, guys, we've reached the end of our logarithmic journey! We started with the equation ${\log _2 (3t+1) = 16}$ and not only solved it step-by-step but also explored the fascinating world of logarithms and their real-world applications. You've learned the importance of understanding the basics, avoiding common mistakes, and appreciating the versatility of these mathematical tools.

Solving logarithmic equations might seem daunting at first, but with a solid grasp of the fundamentals and a bit of practice, you can conquer any log that comes your way. Remember the key steps:

1. Convert the logarithmic equation to exponential form.
2. Use algebraic techniques to isolate the variable.
3. Check your solutions to avoid extraneous solutions.

Logarithms are more than just abstract mathematical concepts; they are powerful tools that help us understand and quantify the world around us. From measuring earthquakes to analyzing sound intensity, logarithms are essential in many fields.

So, keep practicing, keep exploring, and keep those logarithmic skills sharp. You never know when they might come in handy! And remember, math is not just about finding the right answer; it's about the journey of discovery and the satisfaction of solving a challenging problem. Keep that curiosity alive, and you'll go far!