Solving Logarithmic Equations: A Step-by-Step Guide

Nov 10, 2025 by ADMIN 52 views

Hey math enthusiasts! Ever stumbled upon an equation with a logarithm and felt a little lost? Don't worry, it happens to the best of us! Today, we're diving deep into the world of solving logarithmic equations, with a specific focus on the equation: $7 \log _4(3 w+1)+4=25$ . We'll break it down step by step, making sure you grasp every concept along the way. Get ready to flex those math muscles and become a logarithm-solving pro! We will use markdown to present the solution, ensuring clarity and ease of understanding for everyone.

Understanding Logarithmic Equations: The Basics

Before we jump into the nitty-gritty of solving our specific equation, let's brush up on the fundamentals of logarithmic equations. At its core, a logarithmic equation is simply an equation that involves a logarithm. A logarithm, in the simplest terms, is the inverse function to exponentiation. It answers the question: "To what power must we raise a base to get a certain number?" For example, in the expression $\log_2 8 = 3$ , the base is 2, the number is 8, and the logarithm (the answer) is 3 because $2^3 = 8$ . Understanding this relationship is crucial. When we solve logarithmic equations, we're essentially trying to find the value(s) of the variable that make the equation true. This often involves manipulating the equation to isolate the logarithm, converting the logarithmic form to exponential form, and then solving for the variable. The key to success here, as with many math problems, is paying close attention to the properties of logarithms and the rules of algebra. Be patient, take your time, and don’t be afraid to double-check your work. You've got this!

Solving logarithmic equations can appear daunting, but it becomes manageable with a systematic approach. The initial step usually involves isolating the logarithmic term. This entails using algebraic manipulations to get the logarithmic expression by itself on one side of the equation. Following isolation, the equation needs to be rewritten using the fundamental relationship between logarithms and exponents. The logarithmic form can be converted into exponential form, simplifying the equation and making it solvable using algebraic methods. It is also important to remember that solutions need to be verified in the original equation. We must do this because extraneous solutions can sometimes be introduced during the solving process. These are solutions that satisfy the transformed equation but not the original one. Therefore, always substitute the obtained values back into the initial equation to ensure they are valid. This step is crucial for ensuring accuracy and reliability in your solutions. Mastering these principles will equip you with a solid foundation for tackling any logarithmic equation that comes your way, so let's start with our equation!

Properties of Logarithms

Before we get to the actual solution, let's quickly review some key properties of logarithms that will be helpful in this process:

Product Rule: $\log_b (xy) = \log_b x + \log_b y$
Quotient Rule: $\log_b (x/y) = \log_b x - \log_b y$
Power Rule: $\log_b (x^p) = p \log_b x$
Change of Base Formula: $\log_b x = \frac{\log_a x}{\log_a b}$

These properties are like secret weapons when it comes to simplifying and solving logarithmic equations. Keeping them in mind will make your life a whole lot easier!

Solving the Equation: $7 \log _4(3 w+1)+4=25$

Alright, let's roll up our sleeves and solve the equation: $7 \log _4(3 w+1)+4=25$ . We will dissect this problem into a series of clear, easy-to-follow steps. By following these steps, you will not only arrive at the correct answer but also gain a deeper understanding of the process involved. Let's get started, shall we?

Step 1: Isolate the Logarithmic Term

The first goal is to isolate the logarithmic term, which in this case is $7 \log _4(3 w+1)$ . We need to get this term by itself on one side of the equation. Here’s how:

Subtract 4 from both sides: $7 \log _4(3 w+1) + 4 - 4 = 25 - 4$ This simplifies to: $7 \log _4(3 w+1) = 21$
Divide both sides by 7: $\frac{7 \log _4(3 w+1)}{7} = \frac{21}{7}$ This gives us: $\log _4(3 w+1) = 3$

Now, we've successfully isolated the logarithmic term. Great job, guys! The next step is to get rid of the logarithm.

Step 2: Convert to Exponential Form

Now that the logarithmic term is isolated, we need to convert the equation from logarithmic form to exponential form. Remember the definition of a logarithm? $\log_b x = y$ is equivalent to $b^y = x$ . Applying this to our equation, $\log _4(3 w+1) = 3$ , we get:

$4^3 = 3w + 1$

See how the base of the logarithm (4) becomes the base of the exponent, and the value on the other side of the equation (3) becomes the exponent? This is a crucial step in solving logarithmic equations. It allows us to remove the logarithm and work with a standard algebraic equation.

Step 3: Solve for w

Now we have a simple algebraic equation to solve. Let's do it:

Calculate 4^3: $4^3 = 64$ So, the equation becomes: $64 = 3w + 1$
Subtract 1 from both sides: $64 - 1 = 3w + 1 - 1$ This gives us: $63 = 3w$
Divide both sides by 3: $\frac{63}{3} = \frac{3w}{3}$ Which results in: $w = 21$

We've found a potential solution, $w = 21$ . But are we done? Not quite!

Step 4: Verify the Solution

It’s super important to verify our solution by plugging it back into the original equation to make sure it works and isn't an extraneous solution. Let's substitute $w = 21$ into the original equation: $7 \log _4(3 w+1)+4=25$

Substitute w with 21: $7 \log _4(3(21)+1)+4=25$
Simplify: $7 \log _4(63+1)+4=25$ $7 \log _4(64)+4=25$
Evaluate the logarithm: Since $4^3 = 64$ , then $\log _4(64) = 3$ So, the equation becomes: $7(3) + 4 = 25$
Simplify and check: $21 + 4 = 25$ $25 = 25$

It checks out! Our solution, $w = 21$ , is indeed valid. Always verify your answers, folks. It saves a lot of headaches in the long run.

Conclusion: The Solution Set

So, after all that hard work, what's our final answer? The solution set for the equation $7 \log _4(3 w+1)+4=25$ is simply {21}. Congrats, you've successfully solved a logarithmic equation! Remember, the key is to isolate the logarithmic term, convert to exponential form, solve for the variable, and always verify your answer. Keep practicing, and you'll become a logarithmic equation master in no time! Remember the steps: isolate, convert, solve, verify. That's your mantra for conquering logarithmic equations. Keep practicing, and you'll be acing these problems in no time! If you feel confident in your skills, try different problems with different bases or expressions inside the logarithm. Happy solving!

This step-by-step guide is designed to not just show you how to solve the equation, but also why each step is taken. Understanding the logic behind the math makes it easier to tackle similar problems in the future. Go forth and conquer those logarithmic equations! Remember to practice, stay curious, and never be afraid to ask for help. Keep learning, keep growing, and keep exploring the amazing world of mathematics!