Problem 29 - Calculus

Question

Find the inverse function. $$p(t)=(1.04)^{t}$$

Step-by-Step Solution

Answer

The inverse function is \(p^{-1}(t) = \frac{\ln(t)}{\ln(1.04)}\).

Step 1: Understand the problem

We need to find the inverse of the function \(p(t) = (1.04)^t\). This means finding a function \(t\) in terms of \(p\) such that \(p = (1.04)^t\).

Step 2: Swap the variables

To find the inverse, start by swapping the variables \(p\) and \(t\). Hence, the equation becomes \(t = (1.04)^p\).

Step 3: Isolate the exponent

The goal is to express \(p\) in terms of \(t\). To do this, take the natural logarithm of both sides of the equation: \(\ln(t) = \ln((1.04)^p)\).

Step 4: Use logarithm properties

Apply the power rule of logarithms to rewrite the equation: \(\ln(t) = p \cdot \ln(1.04)\).

Step 5: Solve for the inverse

Isolate \(p\) by dividing both sides by \(\ln(1.04)\): \(p = \frac{\ln(t)}{\ln(1.04)}\). This is the inverse function.

Key Concepts

Exponential Functions

Exponential functions are mathematical functions of the form \( f(x) = a^x \), where \( a \) is a constant greater than zero and \( x \) is the exponent. These functions are characterized by their rate of growth or decay, which increases rapidly as \( x \) becomes larger. They are used to model a variety of real-world scenarios, such as population growth, radioactive decay, and interest calculations. In the exponential function \( p(t) = (1.04)^t \), \( 1.04 \) is the base, representing a growth factor, meaning the function models something that increases by 4% per unit \( t \).

Some key features of exponential functions include:

• They are defined for all real numbers \( x \).
• Their graphs pass through the point \((0,1)\) because any non-zero number to the power of zero is 1.
• They are continuous and smooth, with no sharp points or breaks.

Logarithms

Logarithms are the inverse of exponential functions and answer the question: "To what power must we raise a base number to get another number?" For any base \( a \), the logarithm of \( x \) is denoted as \( \log_a(x) \). This means \( a^y = x \) is equivalent to \( y = \log_a(x) \).

Key properties of logarithms include:

• Product Rule: \( \log_a(xy) = \log_a(x) + \log_a(y) \)
• Quotient Rule: \( \log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y) \)
• Power Rule: \( \log_a(x^b) = b \cdot \log_a(x) \)

In solving for the inverse of \( p(t) = (1.04)^t \), logarithms come into play as we convert the relationship into a form that allows us to isolate the variable of interest.

Solving Equations

When solving equations, the aim is to find the value of the unknown variable that makes the equation true. Steps often involve rearranging the equation, factoring, or using operations like addition, subtraction, multiplication, division, or taking roots. In the context of finding an inverse function, solving the equation involves expressing the output variable (originally the dependent variable) back in terms of the independent variable.

For example, in the equation \( t = (1.04)^p \), we take the following approach:

• Understand what the equation represents and what you need to find.
• Use appropriate mathematical operations like taking logarithms to simplify and rearrange the equation.
• Isolate the variable of interest, ensuring all operations are reversible.

This methodical approach leads to the successful derivation of the inverse function \( p = \frac{\ln(t)}{\ln(1.04)} \).

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the logarithm to the base \( e \), where \( e \) (approximately equal to 2.718) is a mathematical constant. It is particularly important in calculus and in modelling natural growth processes because of its unique properties that make differentiation and integration more straightforward.

When we take the natural logarithm of both sides in an equation, it allows us to bring down exponents and solve for variables with ease. In our exercise, applying \( \ln \) to both sides of the equation \( t = (1.04)^p \) transformed it into \( \ln(t) = p \cdot \ln(1.04) \).

Natural logarithms are widely used for:

• Simplifying calculations involving exponentials in financial calculations (like compound interest).
• Analyzing natural growth and decay processes.
• Supporting the solving of equations where \( e^x \) is involved.

Mastering the use of natural logarithms is a robust skill, especially when dealing with continuously compounding scenarios.