Problem 144 - Algebra and Trigonometry

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

The first two statements are true; the third is false. The corrected statement could be \(10^{x}=5.71\) with 'x' now in the exponent.

Step 1: Analysis of first equation

Looking at the first equation \(10^{x}=5.71\), it is easily seen that it is in exponential form, with 10 as the base and x as the exponent. Therefore, this is a true statement.

Step 2: Analysis of second equation

The second equation \(e^{x}=0.72\) is also in exponential form, but with 'e' (the base of the natural logarithm, approximately equal to 2.71828) as the base and x as the exponent. Hence this is also a true statement.

Step 3: Analysis of third equation

Examining the final equation \(x^{10}=5.71\), we find that the variable 'x' is the base and 10 is the exponent, rather being in the exponent itself. This does not match the standard form \(a^{x}\) for exponential equations, hence this is a false statement.

Step 4: Correction of false statement

The correct form of the equation should have 'x' in the exponent. An example of a correct form could be \(10^{x}=5.71\) where '10' is the base and 'x' is the exponent.