Potential Solutions For Logarithmic Equation 2log₅x = Log₅4
In the realm of mathematics, particularly when dealing with logarithmic equations, finding the correct solutions requires a deep understanding of the properties and rules governing logarithms. This article will delve into the step-by-step process of solving the equation 2log₅x = log₅4, exploring potential solutions, and verifying their validity within the domain of logarithmic functions. Our goal is to provide a comprehensive guide that not only answers the specific question but also enhances your overall grasp of logarithmic equations.
Understanding Logarithmic Equations
Before we dive into the solution, let's briefly review what logarithmic equations are and why certain values might or might not be valid solutions. A logarithmic equation is an equation that involves logarithms of expressions containing a variable. The logarithm of a number x with respect to a base b (written as log_b(x)) represents the exponent to which b must be raised to produce x. The key here is that the argument of a logarithm (the x in log_b(x)) must be strictly positive. This is because logarithms are only defined for positive numbers, a critical point to remember when solving logarithmic equations.
When solving these equations, it’s not enough to just manipulate and find potential values for the variable. We must also check these values against the original equation to ensure they do not result in taking the logarithm of a non-positive number. This process of verification is crucial in logarithmic equations, where potential solutions derived algebraically may not actually be valid due to the domain restriction of logarithmic functions.
Step-by-Step Solution of 2log₅x = log₅4
Now, let's systematically solve the given equation, 2log₅x = log₅4. We will employ the properties of logarithms to simplify the equation, isolate the variable, and then verify our solutions. This step-by-step approach will illustrate the correct methodology for tackling such problems.
Step 1: Applying the Power Rule of Logarithms
The first step involves using the power rule of logarithms, which states that log_b(x^p) = plog_b(x)*. This rule allows us to simplify the left side of the equation by moving the coefficient 2 in front of the logarithm as an exponent of x. Applying this rule to our equation, we get:
log₅(x²) = log₅(4)
Step 2: Equating the Arguments
Now that we have a single logarithm on each side of the equation with the same base (5), we can equate the arguments. This means that if log_b(A) = log_b(B), then A = B. Applying this principle, we set the arguments of the logarithms equal to each other:
x² = 4
Step 3: Solving the Quadratic Equation
We now have a simple quadratic equation to solve. To find the values of x that satisfy this equation, we take the square root of both sides:
√(x²) = ±√4
x = ±2
This gives us two potential solutions: x = 2 and x = -2. It is at this stage that many might be tempted to conclude that we have found our solutions. However, we must proceed with caution and remember the domain restrictions of logarithmic functions.
Step 4: Verifying the Solutions
The final, and arguably most critical, step is to verify whether these potential solutions are valid by substituting them back into the original equation. Remember, we cannot take the logarithm of a negative number or zero, so we must check if our solutions result in taking the logarithm of a positive number.
Checking x = 2
Substitute x = 2 into the original equation:
2log₅(2) = log₅(4)
Using the power rule in reverse, we get:
log₅(2²) = log₅(4)
log₅(4) = log₅(4)
This is a true statement, so x = 2 is a valid solution.
Checking x = -2
Substitute x = -2 into the original equation:
2log₅(-2) = log₅(4)
Here, we encounter a problem. We are trying to take the logarithm of a negative number, which is undefined in the real number system. Therefore, x = -2 is not a valid solution.
Conclusion: The Valid Solution
After carefully solving the equation 2log₅x = log₅4 and verifying the potential solutions, we find that only one solution is valid: x = 2. The value x = -2 is extraneous because it results in taking the logarithm of a negative number, which is not allowed. This exercise underscores the importance of not only understanding the properties of logarithms but also the necessity of verifying solutions in the context of the domain of logarithmic functions. When dealing with logarithms, always remember to check your answers to ensure they are valid within the function's domain.
In the realm of logarithmic equations, accurately identifying potential solutions is a crucial skill for any mathematics enthusiast. This article zeroes in on the equation 2log₅x = log₅4, dissecting the process of determining which of the provided options—x = -2, x = 4, x = 16, x = -10, and x = 2—are indeed potential solutions. We’ll navigate through the properties of logarithms, the significance of domain restrictions, and the verification process to ensure a comprehensive understanding.
Reviewing the Fundamentals of Logarithmic Equations
Before we plunge into the specifics of the equation at hand, let's solidify our grasp on logarithmic equations in general. A logarithmic equation is, at its core, an equation that involves the logarithm of an expression containing a variable. The logarithm of a number x to a base b, denoted as log_b(x), essentially asks the question:
