Solving Logarithmic Equations: A Step-by-Step Guide

Are you struggling with logarithmic equations? Don't worry, you're not alone! Logarithmic equations can seem daunting at first, but with a clear understanding of the rules and properties of logarithms, they can be solved systematically. This guide will walk you through the process of solving the equation log(20x³) - 2log(x) = 4, providing a detailed explanation of each step along the way. We'll break down the equation, apply logarithmic properties, and ultimately find the solution for x. So, let's dive in and conquer those logarithms together!

Understanding Logarithms

Before we tackle the equation, let's refresh our understanding of logarithms. A logarithm is essentially the inverse operation of exponentiation. In simple terms, if we have an equation like b^y = x, then the logarithm (base b) of x is y, written as log_b(x) = y. The base 'b' is the number that is raised to a power. When no base is written (like in our equation), it is assumed to be base 10. Understanding this fundamental relationship between exponents and logarithms is crucial for solving logarithmic equations. Key properties of logarithms include:

• Product Rule: log_b(mn) = log_b(m) + log_b(n)
• Quotient Rule: log_b(m/n) = log_b(m) - log_b(n)
• Power Rule: log_b(m^p) = p * log_b(m)

These properties will be our tools as we unravel the equation. These logarithmic identities act as the fundamental rules governing how we manipulate and simplify expressions involving logarithms. The product rule allows us to break down the logarithm of a product into the sum of individual logarithms. The quotient rule, conversely, lets us express the logarithm of a quotient as the difference between logarithms. And the power rule is particularly useful when dealing with exponents within logarithms, as it enables us to bring the exponent down as a coefficient. Mastering these properties is paramount, as they provide the means to consolidate logarithmic terms, eliminate complexities, and ultimately transform the equation into a form that can be readily solved. By skillfully applying these rules, we can manipulate logarithmic expressions to isolate the variable and determine its value.

The Problem: log(20x³) - 2log(x) = 4

Now, let's revisit our equation: log(20x³) - 2log(x) = 4. Our goal is to isolate 'x' and find its value. To do this, we'll use the logarithmic properties we just discussed. The first step involves applying the power rule to the second term, 2log(x). This rule allows us to rewrite the term as log(x²). Applying the power rule to simplify the equation is a crucial strategic move. This initial manipulation streamlines the equation by consolidating coefficients and making the logarithmic terms more amenable to further operations. The power rule, which states that log_b(m^p) = p * log_b(m), provides a direct means to address exponents within logarithms. In our specific case, 2log(x) can be transformed into log(x²) using this rule. This transformation is not merely cosmetic; it sets the stage for subsequent steps where we can leverage other logarithmic properties, such as the quotient rule or the product rule, to combine terms and simplify the equation further. By skillfully applying the power rule, we lay the groundwork for a more efficient and effective solution process.

Our equation now looks like this: log(20x³) - log(x²) = 4

Applying the Quotient Rule

Next, we can use the quotient rule of logarithms, which states that log_b(m/n) = log_b(m) - log_b(n). Applying this rule to our equation, we can combine the two logarithmic terms into a single logarithm. The quotient rule of logarithms acts as a bridge, connecting the difference of two logarithms to the logarithm of a single quotient. This property is indispensable when we aim to consolidate multiple logarithmic terms into a more manageable form. In our equation, where we have log(20x³) - log(x²), the quotient rule provides the perfect tool for combining these terms. By recognizing the inherent structure of the equation, we can apply the rule to transform the expression into a single logarithm: log((20x³)/(x²)). This consolidation simplifies the equation significantly, making it easier to manipulate and solve. The quotient rule, therefore, is not just a mathematical formula; it's a strategic asset that empowers us to streamline complex expressions and pave the way for a more efficient solution.

This gives us: log((20x³)/(x²)) = 4

Simplifying the fraction inside the logarithm, we get: log(20x) = 4

Converting to Exponential Form

Now, we need to get rid of the logarithm. To do this, we convert the equation from logarithmic form to exponential form. Remember that log_b(x) = y is equivalent to b^y = x. Since our logarithm is base 10 (because no base is written), we can rewrite the equation as. Transforming the equation from logarithmic to exponential form is a pivotal step in unraveling the solution. This conversion effectively liberates the variable from the confines of the logarithm, allowing us to work with a more straightforward algebraic expression. The fundamental relationship between logarithms and exponents is the key to this transformation: if log_b(x) = y, then b^y = x. In our case, where we have log(20x) = 4 (with the base understood to be 10), this relationship dictates that we can rewrite the equation as 10^4 = 20x. This transformation not only eliminates the logarithm but also restructures the equation into a form that is more amenable to algebraic manipulation. By understanding and applying this principle, we gain the ability to bridge the gap between logarithmic and exponential expressions, opening up new avenues for solving complex equations.

10^4 = 20x

Solving for x

10^4 is equal to 10,000, so our equation is now: 10,000 = 20x. To isolate x, we divide both sides of the equation by 20. Isolating the variable 'x' is the ultimate objective of solving any equation. In our journey to decipher the value of 'x' in the equation 10,000 = 20x, the final step involves a simple yet crucial algebraic maneuver: dividing both sides of the equation by the coefficient of 'x'. This operation maintains the balance of the equation while effectively separating 'x' from its numerical multiplier. By dividing both 10,000 and 20x by 20, we achieve the isolation we seek. This division unveils the direct relationship between 'x' and its value, allowing us to express 'x' as the quotient of 10,000 and 20. The act of isolating 'x' is not just a mathematical procedure; it's the culmination of our efforts to unravel the equation and reveal the hidden value of the unknown.

This gives us:

x = 10,000 / 20

x = 500

Checking the Solution

It's always a good idea to check our solution by plugging it back into the original equation. Substituting x = 500 into log(20x³) - 2log(x) = 4, we get:

log(20 * 500³) - 2log(500) = 4

This simplifies to:

log(2.5 * 10^10) - 2log(500) = 4

Using a calculator, we can verify that this is approximately equal to 4, so our solution is correct.

Conclusion

We have successfully solved the logarithmic equation log(20x³) - 2log(x) = 4. By understanding the properties of logarithms and applying them systematically, we were able to isolate x and find its value. Remember to always check your solution to ensure accuracy. Logarithmic equations might seem complex, but with practice and a solid understanding of the fundamentals, you can master them. For further learning and practice, you can explore resources like Khan Academy's Logarithm section, which offers detailed explanations and exercises.