Problem 144 - College Algebra Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Examples of exponential equations include \(10^{x}=5.71\) \(e^{x}=0.72,\) and \(x^{10}=5.71\)

Step-by-Step Solution

Answer

Statement 1 is true, Statement 2 is true, Statement 3 is false and can be corrected as \(10^{x}=5.71\) to form an exponential equation.

Step 1: Analyzing Statement 1

The first equation provided is \(10^{x}=5.71\). In this equation, the variable \(x\) is in the exponent which aligns with the definition of an exponential equation. Therefore, this statement is true.

Step 2: Analyzing Statement 2

The second equation provided is \(e^{x}=0.72\). This is an equation in which the variable \(x\) appears in the exponent. Therefore, by definition, this is also an exponential equation and the statement is true.

Step 3: Analyzing Statement 3

The third equation given is \(x^{10}=5.71\). However, in this equation, the variable \(x\) is not in the exponent but in the base. Thus, this statement is a power equation, not an exponential one, making the statement false. To make it true and become an exponential equation, it should be reversed to \(10^{x}=5.71\).