Solving Log_5(x+30) = 3 A Step-by-Step Solution

Introduction

In this comprehensive guide, we will delve into the solution of the logarithmic equation ${\log_5(x+30) = 3}$. Logarithmic equations can seem daunting at first, but with a clear understanding of the properties of logarithms and a systematic approach, they become quite manageable. Our primary goal here is to provide a step-by-step solution that not only answers the question but also enhances your understanding of logarithmic functions and equation-solving techniques. This article is designed for students, educators, and anyone looking to brush up on their algebra skills. We aim to make the process as clear and intuitive as possible, ensuring that you can tackle similar problems with confidence in the future.

Understanding Logarithms

Before we dive into solving the equation ${\log_5(x+30) = 3}$, it's crucial to understand the fundamental concept of logarithms. A logarithm is essentially the inverse operation to exponentiation. In simple terms, if we have an exponential equation like ${5^3 = 125}$, the logarithmic equivalent is ${\log_5(125) = 3}$. Here, 5 is the base, 125 is the argument, and 3 is the exponent. Understanding this relationship is the cornerstone of solving logarithmic equations. The general form of a logarithmic equation is ${\log_b(a) = c}$, where ${b}$ is the base, ${a}$ is the argument, and ${c}$ is the result. This equation can be rewritten in exponential form as ${b^c = a}$. This conversion between logarithmic and exponential forms is the key to unraveling many logarithmic problems. It's also essential to remember that the base ${b}$ must be a positive number not equal to 1, and the argument ${a}$ must be positive. These constraints are critical for the logarithm to be defined. By grasping these basics, you'll find solving logarithmic equations much less intimidating.

Properties of Logarithms

To further equip you in solving logarithmic equations, let's briefly discuss some essential properties of logarithms. These properties provide tools to manipulate and simplify logarithmic expressions, making complex equations more tractable. One fundamental property is the product rule, which states that ${\log_b(mn) = \log_b(m) + \log_b(n)}$. This means the logarithm of a product is the sum of the logarithms of the individual factors. Another crucial property is the quotient rule, which asserts that ${\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n)}$. This rule indicates that the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator. The power rule is another key property, expressing that ${\log_b(m^p) = p \log_b(m)}$. This rule allows us to bring exponents outside the logarithm as a multiplicative factor. Additionally, we have the change of base formula, which is particularly useful when dealing with logarithms of different bases. It states that ${\log_b(a) = \frac{\log_c(a)}{\log_c(b)}}$, where ${c}$ is a new base. Finally, it's worth noting the identities ${\log_b(1) = 0}$ and ${\log_b(b) = 1}$, which are frequently used in simplifying expressions. By mastering these properties, you'll be well-prepared to tackle a wide array of logarithmic equations and problems.

Step-by-Step Solution

Now, let’s solve the equation ${\log_5(x+30) = 3}$ step by step. This process will illustrate how to apply the principles of logarithms to find the value of ${x}$. Our primary focus will be on converting the logarithmic equation into its equivalent exponential form, which is a crucial step in solving such problems. We will then isolate ${x}$ to find the solution. Each step is meticulously explained to ensure clarity and understanding.

Step 1: Convert to Exponential Form

The first and most crucial step in solving the logarithmic equation ${\log_5(x+30) = 3}$ is to convert it into its equivalent exponential form. Recall that a logarithmic equation of the form ${\log_b(a) = c}$ can be rewritten as ${b^c = a}$. Applying this to our equation, where ${b = 5}$, ${a = x+30}$, and ${c = 3}$, we can rewrite ${\log_5(x+30) = 3}$ as ${5^3 = x+30}$. This transformation is the key to unlocking the problem, as it allows us to work with a more familiar algebraic structure. The exponential form eliminates the logarithm, making it easier to isolate ${x}$ and find its value. This step highlights the fundamental relationship between logarithms and exponentials, demonstrating how they are inverse operations of each other. By making this conversion, we've set the stage for the subsequent steps in the solution process.

Step 2: Simplify the Exponential Term

Following the conversion of the logarithmic equation ${\log_5(x+30) = 3}$ to its exponential form ${5^3 = x+30}$, the next step is to simplify the exponential term. Calculating ${5^3}$ means multiplying 5 by itself three times: ${5 \times 5 \times 5}$. This equals 125. So, our equation now becomes ${125 = x+30}$. Simplifying the exponential term makes the equation more straightforward and easier to solve. This step is a fundamental arithmetic operation but is crucial for progressing towards the solution. By reducing the exponential term to a simple numerical value, we bring the equation closer to a form where we can isolate the variable ${x}$. This simplification is a common technique in solving various types of equations and helps in making the problem more manageable.

Step 3: Isolate the Variable ${x}$

After simplifying the exponential term in the equation ${125 = x+30}$, the next crucial step is to isolate the variable ${x}$. To do this, we need to eliminate the constant term on the same side of the equation as ${x}$. In this case, we have ${+30}$ added to ${x}$. To isolate ${x}$, we perform the inverse operation, which is subtraction. We subtract 30 from both sides of the equation to maintain balance and equality. This gives us ${125 - 30 = x + 30 - 30}$. Simplifying both sides, we get ${95 = x}$. This process of isolating the variable is a fundamental algebraic technique used in solving various types of equations. By performing the same operation on both sides, we ensure that the equation remains balanced and that the value of ${x}$ is accurately determined. This step brings us to the solution of the equation, where we have successfully isolated ${x}$ and found its value.

Step 4: State the Solution

Having isolated the variable ${x}$ in the equation and found that ${x = 95}$, the final step is to explicitly state the solution. This is a critical part of the problem-solving process, as it clearly communicates the answer we have found. In this case, the solution to the logarithmic equation ${\log_5(x+30) = 3}$ is ${x = 95}$. Stating the solution clearly leaves no ambiguity and provides a definitive answer to the problem. It also allows for easy verification, as we can substitute this value back into the original equation to check if it holds true. For example, plugging ${x = 95}$ back into the original equation gives us ${\log_5(95+30) = \log_5(125)}$, which simplifies to ${\log_5(5^3) = 3}$, thus confirming our solution. This final step ensures that the problem is completely solved and the answer is clearly presented.

Verification

To ensure the accuracy of our solution, we must verify it. This step is crucial in solving logarithmic equations as it confirms that the value we found for ${x}$ satisfies the original equation. Verification involves substituting the obtained value of ${x}$ back into the original equation and checking if the equation holds true. This process not only validates our solution but also helps in identifying any potential errors made during the solving process. By performing this check, we can have confidence in the correctness of our answer. Let’s proceed with the verification process for the equation we solved.

Substituting ${x = 95}$ into the Original Equation

To verify our solution ${x = 95}$ for the logarithmic equation ${\log_5(x+30) = 3}$, we substitute 95 in place of ${x}$ in the original equation. This gives us ${\log_5(95+30) = 3}$. The next step is to simplify the expression inside the logarithm. Adding 95 and 30, we get 125. So, our equation now looks like ${\log_5(125) = 3}$. This substitution allows us to check whether the left-hand side of the equation equals the right-hand side, thus confirming the validity of our solution. By carefully performing this substitution, we ensure that we are accurately testing whether the value of ${x}$ we found is indeed the correct solution to the given logarithmic equation. This process is a fundamental step in solving mathematical problems and helps in building confidence in our results.

Checking the Equality

After substituting ${x = 95}$ into the original logarithmic equation, we arrived at ${\log_5(125) = 3}$. Now, we need to check if this equality holds true. Recall that ${125}$ can be expressed as ${5^3}$. Therefore, we can rewrite the equation as ${\log_5(5^3) = 3}$. Using the property of logarithms that ${\log_b(b^c) = c}$, we can simplify the left-hand side of the equation. In this case, ${\log_5(5^3)}$ simplifies to 3. So, we now have ${3 = 3}$, which is a true statement. This confirms that our solution, ${x = 95}$, is correct. The process of checking the equality is a critical step in verifying solutions to logarithmic equations, as it ensures that the value we found for ${x}$ satisfies the original equation. By verifying the equality, we can confidently conclude that our solution is accurate and that we have correctly solved the problem.

Conclusion

In summary, we have successfully solved the logarithmic equation ${\log_5(x+30) = 3}$ by following a systematic approach. We began by converting the logarithmic equation into its equivalent exponential form, which allowed us to simplify the problem. We then simplified the exponential term, isolated the variable ${x}$, and found that ${x = 95}$. Finally, we verified our solution by substituting it back into the original equation and confirming that it holds true. This step-by-step process demonstrates the importance of understanding the relationship between logarithms and exponentials, as well as the application of fundamental algebraic techniques. By mastering these concepts and techniques, you can confidently tackle a wide range of logarithmic equations. Remember to always verify your solutions to ensure accuracy. This comprehensive guide aims to equip you with the knowledge and skills necessary to solve similar problems in the future.

Final Answer

The solution to the equation ${\log_5(x+30) = 3}$ is:

$$x = 95$$