Solving Logarithmic Equations A Step By Step Guide To Log 7 + Log(x-4) = 1

Introduction

In this article, we will delve into the process of solving a logarithmic equation. Specifically, we will address the equation log 7 + log(x - 4) = 1. Logarithmic equations, often perceived as challenging, become manageable with a clear understanding of logarithmic properties and algebraic manipulation techniques. Our approach will be methodical, breaking down each step to ensure clarity and comprehension. We aim to not only provide the solution but also to equip you with the knowledge to tackle similar problems independently. This exploration will cover key concepts such as the properties of logarithms, the definition of logarithms, and how to convert logarithmic equations into exponential forms. Understanding these fundamentals is crucial for anyone seeking to master algebra and calculus. By the end of this discussion, you will gain a solid understanding of how to approach and solve logarithmic equations, enhancing your mathematical problem-solving skills.

Understanding Logarithms

Before we dive into solving the equation, it's essential to grasp the fundamental concept of logarithms. A logarithm is essentially the inverse operation to exponentiation. In simple terms, if we have an exponential expression like b^y = x, the logarithm answers the question: "To what power must we raise the base 'b' to get 'x'?" This is written as log_b(x) = y. Here, 'b' is the base of the logarithm, 'x' is the argument, and 'y' is the logarithm itself. The equation log_b(x) = y is read as "the logarithm of x to the base b is y". A common type of logarithm is the common logarithm, which has a base of 10. When the base is not explicitly written, it is generally understood to be 10. For example, log(x) is the same as log_10(x). Understanding this notation is crucial for solving logarithmic equations. The properties of logarithms, which we will discuss later, are derived from the properties of exponents. Therefore, a firm grasp of exponential functions will greatly aid in understanding logarithmic functions. Logarithms are not just abstract mathematical concepts; they have real-world applications in various fields, including physics, engineering, and computer science. They are used to model phenomena that span several orders of magnitude, such as the Richter scale for earthquakes and the pH scale for acidity.

Key Properties of Logarithms

To effectively solve logarithmic equations, you need to be familiar with the fundamental properties of logarithms. These properties allow us to manipulate logarithmic expressions and simplify equations. Let's discuss some of the most important ones:

1. Product Rule: The logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this is expressed as log_b(mn) = log_b(m) + log_b(n). This property is particularly useful for combining multiple logarithmic terms into a single term, which can simplify the equation. For instance, if you have an expression like log(2x) + log(3), you can combine it into log(2x * 3) = log(6x). The product rule stems directly from the properties of exponents, where multiplying exponential terms with the same base results in adding the exponents.

2. Quotient Rule: The logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. This is represented as log_b(m/n) = log_b(m) - log_b(n). The quotient rule is the counterpart to the product rule and is used when dealing with division within a logarithmic expression. For example, log(x/2) can be rewritten as log(x) - log(2). This rule is derived from the fact that dividing exponential terms with the same base results in subtracting the exponents.

3. Power Rule: The logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number. The formula is log_b(m^p) = p * log_b(m). The power rule is essential for simplifying logarithms of exponential expressions. If you encounter log(x^3), you can rewrite it as 3 * log(x). This rule is a direct consequence of the power of a power rule in exponents, where (b^m)^n = b^(m*n).

4. Change of Base Rule: This rule allows you to convert logarithms from one base to another. It is especially useful when dealing with calculators that only have common logarithm (base 10) or natural logarithm (base e) functions. The formula is log_b(a) = log_c(a) / log_c(b), where 'c' is any other base. For instance, if you need to calculate log_2(7) and your calculator only has a base 10 logarithm function, you can use the change of base rule to rewrite it as log(7) / log(2). This rule is vital for evaluating logarithms with bases other than 10 or e.

5. Logarithm of the Base: The logarithm of the base itself is always equal to 1. This can be expressed as log_b(b) = 1. This is a straightforward property that follows directly from the definition of a logarithm. Since b^1 = b, the logarithm of b to the base b is 1.

6. Logarithm of 1: The logarithm of 1 to any base is always equal to 0. This is written as log_b(1) = 0. This property is based on the fact that any number raised to the power of 0 is 1 (b^0 = 1).

Understanding and applying these properties correctly is crucial for simplifying and solving logarithmic equations. In the next section, we will use these properties to solve the given equation.

Solving the Equation: log 7 + log(x - 4) = 1

Now, let's apply our knowledge of logarithms to solve the given equation: log 7 + log(x - 4) = 1. Our strategy will be to use the properties of logarithms to combine the logarithmic terms, then convert the equation into exponential form, and finally, solve for x. Here's a step-by-step breakdown:

Step 1: Combine Logarithmic Terms

The first step is to use the product rule of logarithms to combine the two logarithmic terms on the left side of the equation. Recall that log_b(m) + log_b(n) = log_b(mn). Applying this rule, we get:

log 7 + log(x - 4) = log(7 * (x - 4))

So our equation becomes:

log(7(x - 4)) = 1

This step simplifies the equation by reducing the number of logarithmic terms, making it easier to work with.

Step 2: Convert to Exponential Form

Since we are dealing with common logarithms (base 10), we can rewrite the equation in exponential form. The equation log_b(x) = y is equivalent to b^y = x. In our case, the base is 10, so the equation log(7(x - 4)) = 1 can be rewritten as:

10^1 = 7(x - 4)

This conversion is a crucial step in solving logarithmic equations as it eliminates the logarithm and allows us to work with a simpler algebraic equation.

Step 3: Simplify and Solve for x

Now we have a simple algebraic equation to solve. First, simplify the equation:

10 = 7(x - 4)

Next, distribute the 7:

10 = 7x - 28

Add 28 to both sides:

10 + 28 = 7x

38 = 7x

Finally, divide both sides by 7 to solve for x:

x = 38 / 7

So, the solution is x = 38/7.

Step 4: Check for Extraneous Solutions

It's important to check our solution in the original equation to ensure it is valid. Logarithmic functions are only defined for positive arguments. Therefore, we need to make sure that x - 4 > 0. Let's check:

38/7 - 4 > 0

38/7 - 28/7 > 0

10/7 > 0

Since 10/7 is indeed greater than 0, our solution is valid.

Therefore, the solution to the equation log 7 + log(x - 4) = 1 is x = 38/7.

Conclusion

In this article, we successfully solved the logarithmic equation log 7 + log(x - 4) = 1. We began by revisiting the fundamental concepts of logarithms and their properties. Understanding these properties is crucial for manipulating and simplifying logarithmic expressions. We then applied the product rule of logarithms to combine terms, converted the equation to exponential form, and solved for x. Finally, we checked our solution to ensure it was valid. The step-by-step approach we followed not only led us to the solution but also provided a clear and concise method for tackling similar logarithmic equations. Remember, the key to solving logarithmic equations lies in a solid understanding of logarithmic properties and the ability to convert between logarithmic and exponential forms. By mastering these concepts, you can confidently approach and solve a wide range of logarithmic problems. This process reinforces the importance of a methodical approach to problem-solving in mathematics, where each step is logically connected and contributes to the final solution.