Problem 11 - Elementary and Intermediate Algebra

Question

Perform a check to determine whether 4 is a solution of \(\log _{5}(x+1)=2\)

Step-by-Step Solution

Answer

4 is not a solution because it does not satisfy the equation.

Step 1: Understand the Problem

We need to verify whether 4 is a solution of the equation \(\log_{5}(x+1) = 2\). This means we need to check if replacing \(x\) with 4 results in a true statement.

Step 2: Substitute x with 4

Substitute 4 into the equation where \(x\) is: \(\log_{5}(4+1) = \log_{5}(5)\).

Step 3: Simplify the Expression

Calculate \(4 + 1\) to get 5. Our expression is now \(\log_{5}(5)\).

Step 4: Evaluate the Logarithm

Recall that \(\log_{b}(b) = 1\) for any base \(b\). Therefore, \(\log_{5}(5) = 1\).

Step 5: Compare with the Original Equation

The original equation \(\log_{5}(x+1) = 2\) expects the logarithm to evaluate to 2, but \(\log_{5}(5) = 1\). Since 1 is not equal to 2, substituting \(x = 4\) does not satisfy the equation.

Key Concepts

Solution Verification

Verifying the solution to an equation is a crucial part of solving math problems. It ensures that your solution is correct. In this exercise, we're tasked to verify if 4 is a solution to the equation \(\log_{5}(x+1) = 2\). The process of verification involves substituting the potential solution back into the original equation to see if it holds true.

Here's how it works: after substituting \(x = 4\) into the equation, calculate to see if the result of \(\log_{5}(5)\) equals 2. If it does, then 4 is indeed a solution. If not, it isn't. In our case, the simplified form, \(\log_{5}(5)\), equals 1, not 2, indicating that 4 is not a solution. Checking your work like this helps prevent mistakes and solidifies your understanding.

Logarithm Properties

Understanding the properties of logarithms is essential when solving logarithmic equations. They allow you to manipulate and solve equations more easily.

• The property that \(\log_{b}(b) = 1\) is particularly useful. It suggests that the logarithm of a number that matches the base is always 1.
• Another core property is \(\log_{b}(x^n) = n \times \log_{b}(x)\), which is useful for solving equations involving powers.
• Remembering that \(\log_{b}(xy) = \log_{b}(x) + \log_{b}(y)\) and \(\log_{b}(x/y) = \log_{b}(x) - \log_{b}(y)\) can make simplifying expressions more straightforward.

Using these properties, we were able to identify that because \(\log_{5}(5) = 1\), the statement \(\log_{5}(5) = 2\) is false, confirming our solution verification step.

Mathematical Substitution

Mathematical substitution is a basic yet powerful tool used to transform equations. It involves replacing variables with specific values or expressions to simplify or solve the problem.

In this exercise, substitution means replacing \(x\) with 4 in the equation \(\log_{5}(x+1) = 2\), leading to \(\log_{5}(5)\). This step is intended to test if our potential solution is valid.

Correctly applying substitution can often bring clarity to complex problems. It illuminates how each part of the equation behaves with certain values, guiding you toward the correct solution. Even when the substitution reveals a falsehood, like in our case, it is equally informative. By understanding substitution, you'll be better prepared to tackle a wide variety of algebraic and logarithmic equations.