Problem 93 - College Algebra Essentials

Question

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. \(x \log 10^{x}=x^{2}\)

Step-by-Step Solution

Answer

The equation \(x \log 10^{x}=x^{2}\) is true.

Step 1: Understand the properties of logarithms

Recall that \(\log_{a}{a^{x}} = x\). This is because \(\log_{a}\) and \(a^{x}\) are inverse operations.

Step 2: Apply the properties of logarithms

Applying the property of logarithm to the left side of the equation, it simplifies to \(x \cdot \log_{10}{10^{x}} = x \cdot x = x^{2}\).

Step 3: Evaluate the equation

After simplification, it's observed that the left hand side (LHS) equals the right hand side (RHS). Therefore, the equation \(x \log 10^{x}=x^{2}\) stands to be true.

Key Concepts

Properties of Logarithms

Logarithms are powerful mathematical tools that help in simplifying complex expressions. One key property to remember is:

• \(\log_{a}{a^{x}} = x\), which means any number \(a\) raised to \(x\) and then taken as a logarithm base \(a\), results in \(x\).
• This property is crucial in solving logarithmic equations as it allows us to transform exponential forms into linear ones.

For example, if you see \(\log_{10}{10^{x}}\), you can directly simplify it to \(x\) because of the matching base (10 here) and exponent property. This simple rule helps in reducing expressions and is vital for further operations in algebra.

Inverse Operations

Inverse operations are operations that "undo" each other. In terms of exponentiation and logarithms:

• Exponents and logs are inverses. Specifically, if you exponentiate a number and then take the logarithm of the result, you end up with the original number.
• For instance, \(10^{\log_{10}{x}} = x\), demonstrating how these two operations counterbalance each other.

Recognizing these inverse relationships is critical for manipulating expressions. In our original exercise, we used the inverse property when simplifying \(\log_{10}{10^{x}}\) into \(x\), forming a straight path from complexity to simplicity.

Simplification

Simplification involves reducing complex expressions into more manageable forms. In the context of logarithms:

• By applying the property \(\log_{a}{a^{x}} = x\), we can streamline equations significantly.
• For example, in the problem \(x \cdot \log_{10}{10^{x}}\), we reduced \(\log_{10}{10^{x}}\) to \(x\), leading to the expression \(x \cdot x\).
• This results in \(x^2\), clearly showing that the equation simplifies to match both sides (LHS = RHS).

Simplification is key not only for solving equations but for making calculations more intuitive and straightforward. It helps to reach conclusions faster and understand the mathematical relationship clearly.

Evaluation of Equations

Evaluating equations is the process used to verify if the expressions on both sides of an equation are equal. By simplifying and applying properties:

• We attempt to balance the equation.
• In our context, we started with \(x \cdot \log_{10}{10^{x}} = x^{2}\) and reduced the left-hand side to \(x^{2}\).
• Upon finding that both sides of the equation were equal, we verified that the given equation \(x \log 10^{x}=x^{2}\) is indeed true.

Evaluation is an essential step that helps determine the correctness of a given mathematical statement. Mastering this ensures accuracy in mathematical reasoning and problem-solving.