Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation for \(x.\) $$\log _{5} 5=\log _{5} x$$

Step-by-Step Solution

Answer

The solution to the equation \(\log _{5} 5=\log _{5} x\) is \(x = 5\)

Step 1: Analyze the given equation

Given the equation \(\log _{5} 5=\log _{5} x\), it is confirmed that it is on the same base i.e. 5 on both sides of the equation.

Step 2: Apply the property of loagrithms

One of the key properties of logarithms states: If \(\log _{b} a = \log _{b} c\) then \(a = c\). Applying this property to our equation, we get \(5 = x\)

Step 3: Confirm the solution

On observing closely, it is evident that the equation holds true. Replacing \(x\) with \(5\) in the original equation confirms that the both the sides of the equation are indeed equal, hence the solution is confirmed

Key Concepts

Properties of Logarithms

To fully grasp logarithmic equations, like the one presented in the exercise, it's crucial to understand the properties of logarithms. These properties help simplify complex expressions and solve for unknown variables with greater ease. Here's a simple breakdown of the key properties:

• Product Property: \(\log_{b}(mn) = \log_{b} m + \log_{b} n\)This property shows that the logarithm of a product is equal to the sum of the logarithms of the factors.
• Quotient Property: \(\log_{b}\left(\frac{m}{n}\right) = \log_{b} m - \log_{b} n\)This property states that the logarithm of a quotient is equal to the difference of the logarithms.
• Power Property: \(\log_{b}(m^{n}) = n\log_{b} m\)According to this property, the logarithm of a power is equal to the exponent times the logarithm of the base.
• Equality Property: \(\log_{b} a = \log_{b} c \Rightarrow a = c\)This is the key property used in our original exercise. It states that if two logarithms with the same base are equal, then the arguments are equal.

These properties are foundational and widely used in mathematics to manipulate and solve logarithmic equations.

Solving Equations

Solving logarithmic equations requires careful application of the properties of logarithms. In our exercise, we observed an equation of the form \(\log_{5} 5 = \log_{5} x\), which immediately invites the use of the equality property.To solve such equations systematically, follow these general steps:

• Ensure the Same Base: Check that the logarithms have the same base on both sides. This allows us to apply the equality property effectively.
• Apply Properties: Use the properties of logarithms to simplify the equation if needed. In our example, simplification was not necessary since both sides were already equal by structure.
• Isolate the Variable: Use the equality property, which gives us \(a = c\), to directly equate the arguments when the bases are the same.
• Verify Solution: Always substitute the solution variable back into the original equation to confirm that both sides are indeed equal.

By following these logical steps, solving any logarithmic equation becomes a systematic process, ensuring a correct solution.

Mathematical Proofs

Mathematical proofs are used to establish the truth of equations or propositions by logical reasoning. In the context of our exercise, the proof involves confirming that the given equation holds true by theoretical and practical application.Every solution begins with an analysis:

• Understanding the Problem: Before applying any properties, it's critical to thoroughly understand what the equation is asking. This ensures a clear starting point.
• Logical Reasoning: Apply logical steps using known mathematical principles. In our solution, this involves using the equality property of logarithms to match both sides of the equation.
• Concrete Verification: Mathematical solutions are not complete until the results are tested. This means substituting the answer back to check if it satisfies the equation. For \(x = 5\), substituting this back into \(\log_{5} x\) should logically equal \(\log_{5} 5\), thus providing proof that the solution is indeed correct.

Thus, mathematical proofs not only validate a solution but also ensure confidence in the correctness of the approach, promoting a deeper understanding of the underlying principles.