Problem 66 - Algebra 2

Question

Write true or false for each statement. Justify your answer. \(\log x+\log \left(x^{2}+2\right)=\log \left(x^{3}+2 x\right)\)

Step-by-Step Solution

Answer

The statement is true.

Step 1: Apply Properties of Logarithms

Apply the property of logarithms that the sum of two logarithms is equal to the logarithm of their product: \(\log x + \log(x^2 + 2) = \log[x \times (x^2 + 2)]\), simplifying the right side we get \(\log(x^3 + 2x)\), which matches the right-hand side of the original equation.

Step 2: Confirm the Equality

To confirm, we check whether both sides of the equation are indeed equal. The original equation is \(\log x + \log(x^2 + 2) = \log (x^3 + 2x)\). After applying the logarithm property, we have shown that \(\log x + \log(x^2 + 2) = \log(x^3 + 2x)\). Thus, both sides of the equation match exactly, confirming that the original statement is true.

Key Concepts

Logarithmic Equations

Logarithmic equations are equations that involve logarithms. They often require the application of properties of logarithms to solve or simplify. In our exercise, the equation we are exploring is \( \log x + \log (x^2 + 2) = \log (x^3 + 2x) \). Here, we need to verify whether both sides are actually equal, which often involves applying one of the key logarithmic properties. If the logs on both sides have the same base and argument, then the expressions inside are equal, too. This is crucial to understanding how to work through logarithmic equations effectively.

Simplifying Logarithmic Expressions

To simplify logarithmic expressions, we use the properties of logarithms. A fundamental property is \( \log a + \log b = \log(ab) \). It states that the sum of two logarithms (with the same base) is the logarithm of the product of their arguments.
In the exercise, we had:

• \( \log x + \log (x^2 + 2) = \log (x^3 + 2x) \)

By applying the product property, the left side becomes:

• \( \log [x \times (x^2 + 2)] \)
• Which simplifies further to: \( \log (x^3 + 2x) \)

Both expressions are indeed equivalent, confirming our simplification was correct. Simplifying log expressions like this is essential in solving or verifying logarithmic equations.

Verifying Algebraic Equations

Verifying algebraic equations often involves confirming that both sides of the equation express the same value. In the realm of logarithms, once you simplify one side using properties of logarithms, the next step is to ensure it matches the other side.

In our case:

• Original equation: \( \log x + \log (x^2 + 2) = \log (x^3 + 2x) \)
• After simplification and property application, both sides become: \( \log (x^3 + 2x) \)

Since both sides match, we can confidently assert that the equation is balanced, hence, the statement is "true". Verifying algebraic equations, especially those involving logarithms, is a systematic approach. It builds the foundation for solving complex logarithmic problems.