Question

Give a logarithmic and exponential example of when a student would want to use the one to one property. (One of these needs to be more complex.)

Answer 1

The one-to-one property is used when solving logarithmic and exponential equations. It states that if two logarithms (or exponents) have the same base and result in the same value, then the inputs (or exponents) must be equal.

Step-by-step explanation:

The one-to-one property is used when solving logarithmic and exponential equations. For example, if we have the equation log3(x) = log3(8), we can use the one-to-one property to conclude that x = 8. This is because the logarithmic function is a one-to-one function, meaning that if two logarithms have the same base and result in the same value, then the inputs of the logarithms must be equal. Similarly, in exponential equations, such as 2x = 24, we can use the one-to-one property to find that x = 4 since the base and exponent must be equal.

Answer 2

Logarithmic Example:
Suppose a student is solving for the time it takes for a substance to decay to a certain level using the radioactive decay formula \(N = N_0 e^{-kt}\), where \(N\) is the current quantity, \(N_0\) is the initial quantity, \(k\) is a constant, and \(t\) is time. By taking the natural logarithm of both sides, the student can use the one-to-one property to isolate \(t\).

Exponential Example:
Consider a compound interest problem where a student is calculating the time it takes for an investment to double using the formula \(A = P(1 + r/n)^{nt}\), where \(A\) is the future value, \(P\) is the principal amount, \(r\) is the annual interest rate, \(n\) is the number of times interest is compounded per year, and \(t\) is time in years. By applying the one-to-one property, the student can solve for \(t\) after taking the logarithm base \(n\) on both sides.