Problem 100 - College Algebra

Question

In Exercises \(89-102,\) determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$ \log _{6}[4(x+1)]=\log _{6} 4+\log _{6}(x+1) $$

Step-by-Step Solution

Answer

The given equation is true.

Step 1: Representation of the Logarithmic Identity

The first step is to identify that the right side of the given equation contains two terms and represents a form equivalent to \(\log_b{m} + \log_b{n}\). This can be rewritten as \(\log_b{mn}\) according to the logarithmic identity. It is then possible to rewrite this as \(\log _{6}(4 * (x+1))\).

Step 2: Comparison of Both Sides of The Equation

Having rewritten the right side, the equation looks like this: \(\log _{6}(4 * (x+1)) = \log _{6}[4(x+1)]\). When compared to each other, it can be seen that both sides of the equation are equal.

Step 3: Conclusion

Since both sides of the equation are identically the same after applying the logarithmic identity, the given equation is thus true. There is no need to make changes to the equation for it to be true.

Key Concepts

Logarithmic Identities

Logarithmic identities are essential tools in simplifying and solving equations involving logarithms. One of the most commonly used identities is

• Product Identity: \(\log_b(m) + \log_b(n) = \log_b(mn)\)

This identity helps combine two logarithmic expressions into one by converting the sum of logarithms into a single logarithm of a product.
In our exercise, this identity is employed to simplify the right side of the given equation from \(\log_6 4 + \log_6(x+1)\) to \(\log_6 [4(x+1)]\).
Understanding these identities is crucial, as they allow us to manipulate and transform logarithmic expressions accurately.

Equation Solving

Equation solving is the process of finding the values of variables that satisfy a given mathematical statement. In logarithmic equations, solving typically involves using properties of logarithms to isolate the variable.
In this context, once we've simplified the logarithmic expressions, we can equate the arguments of the logarithms if they have the same base. This simplification step reduces the complexity, making it possible to solve for the variable involved.
For our given equation, after applying the logarithmic identity, both sides of the equation become identical. This equivalence means that there's no further solving needed because the equation is true for all permissible values of the variable. However, if not identical, additional steps would be taken to isolate the variable and solve the equation.

Mathematical Proof

A mathematical proof is a logical argument that verifies the truth of a statement or equation. When determining whether an equation is true or false, a proof offers justification for the conclusion.
In the context of our exercise, the proof involves recognizing the applicable logarithmic identity and using it to transform one side of the equation to match the other. The process follows these steps:

• Identify the form of the expression and the applicable identity.
• Use the identity to rewrite the expression.
• Compare both sides of the equation to check for equality.

Once both sides match, as they do in this equation, the equation is proven to be true. Verifying such facts carefully ensures mathematical accuracy.

Comparison of Expressions

Comparing expressions is a critical skill in mathematics, allowing us to analyze the equivalence or differences between two sides of an equation or mathematical statement.
This involves evaluating each part and applying known identities or operations that reveal their relationship. In our exercise, this comparison is straightforward after applying the logarithmic identity to transform one expression into its equivalent form.
After transforming \(\log_6 4 + \log_6(x+1)\) into \(\log_6[4(x+1)]\), comparing it to the left side \(\log_6[4(x+1)]\) confirms that both are indeed the same. This methodical comparison process ultimately determines the truth of the overall equation.
Developing this skill helps in solving a wide array of mathematical problems where simplification and transformation are key.