Solving Logarithmic Equations: Find X

by Alex Johnson 38 views

Have you ever been stumped by a logarithmic equation? Logarithms might seem intimidating at first, but with a few key properties and a bit of practice, you can tackle them with confidence. In this article, we'll break down a specific logarithmic equation step-by-step, showing you how to solve for the unknown variable. This article will guide you through solving the equation -3 log 5 + log x^2 = log (1/125). We will cover the fundamental logarithmic properties, step-by-step solution of the equation, and provide additional tips for solving similar problems.

Understanding Logarithms

Before we dive into the equation, let's refresh our understanding of logarithms. Logarithms are essentially the inverse operation of exponentiation. If we have an equation like b^y = x, the logarithm of x to the base b is written as log_b(x) = y. In simpler terms, the logarithm answers the question: "To what power must we raise the base (b) to get x?" For instance, because 10^2 = 100, we can say that log_10(100) = 2. When no base is explicitly written, it's generally assumed to be base 10. Thus, log(x) typically means log_10(x).

Key Logarithmic Properties

To effectively solve logarithmic equations, we need to be familiar with several key properties:

  1. Product Rule: log_b(mn) = log_b(m) + log_b(n)
  2. Quotient Rule: log_b(m/n) = log_b(m) - log_b(n)
  3. Power Rule: log_b(m^p) = p * log_b(m)
  4. Change of Base Formula: log_b(a) = log_c(a) / log_c(b)
  5. Log of Base: log_b(b) = 1
  6. Log of 1: log_b(1) = 0

These properties allow us to manipulate logarithmic expressions, simplifying them and ultimately helping us solve equations. The logarithmic properties, especially the power rule, quotient rule, and product rule, play a crucial role in simplifying and solving logarithmic equations. Understanding these properties thoroughly is essential for manipulating expressions and isolating the variable. The power rule, for instance, helps us deal with exponents within logarithms, while the product and quotient rules allow us to combine or separate logarithmic terms. These properties are the foundation upon which we build our solution.

Breaking Down the Equation: -3 log 5 + log x^2 = log (1/125)

Now, let's tackle our equation: -3 log 5 + log x^2 = log (1/125). This equation might look a bit complex, but we can simplify it using the logarithmic properties we just discussed. Our main goal is to isolate 'x,' and we'll do that by strategically applying these properties.

Step-by-Step Solution

  1. Apply the Power Rule: Our first step is to address the coefficients and exponents. Using the power rule (log_b(m^p) = p * log_b(m)), we can rewrite the terms: -3 log 5 becomes log (5^-3) and log x^2 remains as is. So our equation now looks like this: log (5^-3) + log x^2 = log (1/125). The power rule is instrumental here, allowing us to move the coefficient of a logarithm into the exponent of its argument, thus simplifying the expression.
  2. Simplify Constants: Let's simplify 5^-3 and 1/125. We know that 5^-3 is equal to 1/(5^3), which is 1/125. So, we have log (1/125) + log x^2 = log (1/125). Simplifying constants makes the equation cleaner and easier to work with. Recognizing that 5^-3 and 1/125 are equivalent helps streamline the process.
  3. Apply the Product Rule: Now, we can combine the logarithmic terms on the left side using the product rule (log_b(m) + log_b(n) = log_b(mn)). This gives us log [(1/125) * x^2] = log (1/125). The product rule is key to consolidating multiple logarithmic terms into a single one, which brings us closer to isolating 'x'. By combining the logarithmic terms on the left side, we reduce the complexity of the equation.
  4. Equate the Arguments: Since the logarithms on both sides of the equation are equal, their arguments must also be equal. This means (1/125) * x^2 = 1/125. Equating the arguments is a critical step in solving logarithmic equations. If log(A) = log(B), then A must equal B. This allows us to eliminate the logarithms and work with a simpler algebraic equation.
  5. Solve for x^2: To isolate x^2, we can multiply both sides of the equation by 125. This gives us x^2 = 1. Solving for x^2 is a straightforward algebraic step that brings us closer to finding the value(s) of 'x'. By isolating x^2, we prepare for the final step of taking the square root.
  6. Solve for x: Finally, we take the square root of both sides of the equation to solve for x. Remember that taking the square root can result in both positive and negative solutions. Therefore, x = ±1. Taking the square root is the final step in isolating 'x', but it's crucial to remember to consider both positive and negative roots.

Final Solution

Thus, the solutions for x are x = 1 and x = -1. We have successfully solved the equation by systematically applying logarithmic properties and algebraic techniques. Always remember to check your solutions in the original equation to ensure they are valid, as logarithms are not defined for negative numbers or zero. In this case, both x = 1 and x = -1 are valid solutions.

Checking the Solutions

It's always a good practice to check our solutions by plugging them back into the original equation. This ensures that our solutions are valid and that we haven't made any errors in our calculations. Let's check both x = 1 and x = -1:

Checking x = 1

Original equation: -3 log 5 + log x^2 = log (1/125)

Substitute x = 1: -3 log 5 + log (1^2) = log (1/125)

Simplify: -3 log 5 + log 1 = log (1/125)

Since log 1 = 0: -3 log 5 = log (1/125)

Rewrite -3 log 5 as log (5^-3): log (5^-3) = log (1/125)

Simplify 5^-3: log (1/125) = log (1/125)

The equation holds true, so x = 1 is a valid solution.

Checking x = -1

Original equation: -3 log 5 + log x^2 = log (1/125)

Substitute x = -1: -3 log 5 + log ((-1)^2) = log (1/125)

Simplify: -3 log 5 + log 1 = log (1/125)

Since log 1 = 0: -3 log 5 = log (1/125)

Rewrite -3 log 5 as log (5^-3): log (5^-3) = log (1/125)

Simplify 5^-3: log (1/125) = log (1/125)

The equation holds true, so x = -1 is also a valid solution. Checking the solutions is a crucial step to ensure accuracy. It confirms that our algebraic manipulations and application of logarithmic properties have led us to the correct answers.

Tips for Solving Logarithmic Equations

Solving logarithmic equations can become quite intuitive with practice. Here are some additional tips to keep in mind:

  1. Always Check for Extraneous Solutions: Logarithmic functions are not defined for non-positive arguments (i.e., negative numbers and zero). Therefore, always plug your solutions back into the original equation to ensure that they don't result in taking the logarithm of a negative number or zero. This step is essential for identifying and discarding extraneous solutions.
  2. Isolate Logarithmic Terms: Before applying any logarithmic properties, try to isolate the logarithmic terms on one side of the equation. This often simplifies the equation and makes it easier to manipulate. Isolating logarithmic terms is a strategic move that sets the stage for applying logarithmic properties effectively.
  3. Use Logarithmic Properties Strategically: Knowing when and how to apply the product, quotient, and power rules is crucial. Practice will help you recognize which properties will be most useful in different situations. Strategic application of logarithmic properties is the key to efficient problem-solving.
  4. Convert to Exponential Form: If you have a single logarithmic term, converting the equation to exponential form can be a helpful strategy. Remember, log_b(x) = y is equivalent to b^y = x. Converting to exponential form can sometimes simplify the equation, especially when dealing with a single logarithmic term.
  5. Practice Regularly: The more you practice, the more comfortable you'll become with logarithmic equations. Work through a variety of problems to develop your skills and intuition. Regular practice is the cornerstone of mastering logarithmic equations. The more you engage with different types of problems, the more adept you'll become at recognizing patterns and applying the appropriate techniques.

Conclusion

Solving logarithmic equations involves understanding the fundamental properties of logarithms and applying them strategically. By breaking down the equation into manageable steps and using the power rule, product rule, and the concept of equating arguments, we successfully found the solutions for x in the equation -3 log 5 + log x^2 = log (1/125). Remember to always check your solutions to avoid extraneous roots. With practice, you'll become more confident in tackling various logarithmic problems. For further reading and to deepen your understanding of logarithms, you might find helpful resources on websites like Khan Academy's Logarithm Section.