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 = x^2

Understanding logarithmic properties is essential for manipulating equations involving logarithms. In this case, the property that log(a^b) = b * log(a) can be applied to simplify the left side of the equation. This allows us to rewrite log(10^x) as x * log(10), which is a constant, facilitating further analysis of the equation.

Exponential functions are functions of the form f(x) = a^x, where 'a' is a constant. In the context of the given equation, recognizing the behavior of exponential functions helps in understanding how they grow compared to polynomial functions like x^2. This comparison is crucial for determining the truth of the equation across different values of x.

Determining the validity of an equation involves checking if both sides are equal for all values of the variable. In this case, we need to analyze the equation x log(10^x) = x^2 to see if it holds true universally or under specific conditions. If false, identifying necessary changes to make the equation true is a key part of the problem-solving process.