Solve The Logarithmic Equation Log(9x+6)=2

Welcome to our math discussion! Today, we're diving into the fascinating world of logarithms to tackle a specific problem: solving the equation log(9x+6)=2. Logarithmic equations can seem intimidating at first glance, but with a systematic approach and a good understanding of logarithmic properties, they become quite manageable. This particular equation involves a basic logarithmic form that we can convert into an exponential one to find the value of 'x'. Let's break down the steps involved in solving this, ensuring we cover all the essential concepts and potential pitfalls. Our goal is to arrive at a clear, understandable solution that demystifies this type of mathematical problem for everyone. We'll explore the definition of a logarithm, its relationship with exponents, and how to apply these principles to isolate our unknown variable, 'x'. Get ready to sharpen your mathematical skills as we embark on this problem-solving journey together.

Understanding Logarithms and the Equation

Before we can solve log(9x+6)=2, it's crucial to grasp the fundamental concept of logarithms. A logarithm is essentially the inverse operation to exponentiation. In simpler terms, if you have an exponential equation like $b^y = x$, the logarithmic form of this is $\log_b(x) = y$. Here, 'b' is the base of the logarithm, 'x' is the argument (the number you're taking the logarithm of), and 'y' is the exponent to which the base must be raised to get 'x'. In our given equation, log(9x+6)=2, the base of the logarithm is not explicitly written. When this happens, it's understood to be the common logarithm, which has a base of 10. So, the equation can be rewritten as log_10(9x+6)=2. This base-10 understanding is critical because it dictates how we'll convert the logarithmic form into its exponential counterpart. The argument of our logarithm is (9x+6), and the result of the logarithm is 2. This means we are looking for the power to which 10 must be raised to equal (9x+6). Recognizing this relationship is the first major step in confidently approaching the solution. The structure of the equation log(9x+6)=2 presents a clear path forward: transform it into an exponential form, which is typically easier to solve for the variable.

Converting Logarithmic Form to Exponential Form

The key to unlocking the solution for log(9x+6)=2 lies in converting the logarithmic equation into its equivalent exponential form. As we established, log(9x+6)=2 implies log_10(9x+6)=2. Using the definition of a logarithm, where $\log_b(x) = y$ is equivalent to $b^y = x$, we can directly apply this to our equation. Here, the base $b = 10$, the argument $x = (9x+6)$, and the result $y = 2$. Therefore, when we convert log_10(9x+6)=2 to its exponential form, we get $10^2 = (9x+6)$. This transformation is a cornerstone of solving logarithmic equations. It turns a problem involving a logarithm into a simpler algebraic equation that we can solve using standard methods. The power of this conversion cannot be overstated; it's the bridge that connects the logarithmic realm to the familiar territory of algebraic manipulation. Once we have $10^2 = (9x+6)$, the next steps involve straightforward arithmetic and algebra to isolate 'x'. This conversion step is where many students find the most clarity, as it simplifies the problem significantly. It's a direct application of the fundamental relationship between logarithms and exponents, a concept that's vital for mastering this area of mathematics. Remember this conversion rule: log_b(x) = y is the same as $b^y = x$. Apply it diligently, and you'll find solving logarithmic equations becomes much more accessible and less daunting.

Solving the Algebraic Equation

Now that we have successfully converted our logarithmic equation into an exponential one, $10^2 = (9x+6)$, we can proceed to solve for 'x' using basic algebra. The first step is to evaluate the exponential term. We know that $10^2$ means 10 multiplied by itself, which equals 100. So, our equation now becomes $100 = (9x+6)$. This is a simple linear equation. Our goal is to isolate 'x' on one side of the equation. To do this, we first need to get the term containing 'x' by itself. We can achieve this by subtracting 6 from both sides of the equation. So, $100 - 6 = 9x + 6 - 6$, which simplifies to $94 = 9x$. Now, 'x' is being multiplied by 9. To isolate 'x', we need to perform the inverse operation, which is division. We divide both sides of the equation by 9: $94 / 9 = 9x / 9$. This gives us $x = 94/9$. This is our potential solution. It's important to note that in mathematics, especially with logarithms, we often need to check our solutions to ensure they are valid. This brings us to the next crucial step: verifying the solution.

Verifying the Solution

In logarithmic equations, it is absolutely essential to verify the solution we found, $x = 94/9$, in the original equation: log(9x+6)=2. This is because the argument of a logarithm must always be positive. If substituting our value of 'x' results in a non-positive argument (zero or a negative number), then that solution is extraneous and must be discarded. Let's substitute $x = 94/9$ back into the argument (9x+6): $9 * (94/9) + 6$. The 9s in the multiplication cancel out, leaving us with $94 + 6$. This sum equals 100. Now, we plug this back into the original equation: log(100)=2. Since we are dealing with the common logarithm (base 10), we are asking: 'To what power must 10 be raised to get 100?'. We know that $10^2 = 100$. Therefore, log_10(100) = 2. This matches the right side of our original equation. The argument (9x+6) evaluated to 100, which is positive, so our solution is valid. This verification step confirms that $x = 94/9$ is indeed the correct and only solution to the equation log(9x+6)=2. Always remember to check your answers in the original equation, especially when dealing with logarithms or square roots, to avoid extraneous solutions.

Conclusion

We have successfully navigated the process of solving the logarithmic equation log(9x+6)=2. By understanding the fundamental relationship between logarithms and exponents, we converted the logarithmic form into its equivalent exponential form, $10^2 = (9x+6)$. This allowed us to simplify the equation to $100 = 9x + 6$, and through basic algebraic manipulation, we isolated 'x' to find the solution $x = 94/9$. Crucially, we then verified this solution by substituting it back into the original equation. This check confirmed that the argument of the logarithm remained positive, ensuring that our solution is valid and not extraneous. Mastering logarithmic equations like this one requires consistent practice and a firm grasp of the underlying principles. Keep exploring and practicing, and you'll find that these problems become increasingly intuitive. For further exploration into the properties of logarithms and more complex equation solving, I recommend visiting resources like Khan Academy, which offers comprehensive guides and practice problems.