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.
x log 10^x = x²

Verified step by step guidance

1. Recall the logarithm property: $\log \left({10}^x\right)=x$ because the base of the logarithm is 10 and $10^x$ is the argument.
2. Rewrite the left side of the equation using this property: $x\backslash ,\log \left({10}^x\right)=x\backslash ,x=x^2$.
3. Compare the left side and the right side of the original equation: both simplify to $x^2$, so the equation is true for all real values of $x$.
4. Since the equation is true as given, no changes are necessary to make it true.
5. To summarize, the key step is recognizing the logarithm property that $\log \left({10}^x\right)=x$, which simplifies the expression and confirms the equality.

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as log_b(b^x) = x. Understanding these properties allows you to rewrite and evaluate logarithmic expressions accurately, which is essential for verifying equations involving logs.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions involves applying the definition of logarithms and their properties to simplify or compute values. For example, recognizing that log 10^x equals x when the base is 10 helps in simplifying the given equation.

Equation Verification and Manipulation

Verifying an equation requires substituting expressions and simplifying both sides to check equality. If false, algebraic manipulation is used to adjust terms and produce a true statement, ensuring a clear understanding of the relationship between variables.