Problem 86 - Calculus for Biology and Medicine

Question

Show that if \(a>1\), then the function \(y=a^{x}\) can be written in the form \(y=e^{\mu x}\), where \(\mu\) is a positive constant. Write \(\mu\) in terms of \(a\).

Step-by-Step Solution

Answer

 = \ln a, and is positive, confirming \( y = a^x \) as \( y = e^{x} \).

Step 1: Express Exponential Function Using Natural Logarithms

To express a general exponential function as a power of the natural exponential function, recall that any number can be written in terms of the natural number, e: \[ a^{x} = (e^{\ln a})^x = e^{x \ln a} \]This means that the exponential function \( y=a^{x} \) can also be expressed as \( y = e^{\mu x} \), where \( \mu = \ln a \).

Step 2: Identify and Interpret 

Recognize that \( \mu = \ln a \) is the conversion factor from base \( a \) to the base \( e \). In our function, it means \( y = e^{\mu x} \) where \( \mu \) is expressed as \( \ln a \), hence a positive constant because \( a > 1 \).

Step 3: Verify  is Positive

Given \( a > 1 \), and because the natural logarithm function is increasing for positive inputs, \( \ln a > 0 \). This ensures that \( \mu \) is positive, meeting the problem’s requirements.

Key Concepts

Natural Logarithms

Natural logarithms are a type of logarithm where the base is the mathematical constant \( e \), approximately equal to 2.71828. This unique base is called the "natural" base because it often appears in natural growth and decay processes. When we refer to \( \ln a \), we mean the natural logarithm of \( a \), which essentially asks: to what power must we raise \( e \) to obtain \( a \)?
This concept is crucial when converting other exponential forms to base \( e \), making it easier to manipulate and analyze these expressions in calculus and advanced mathematics. Since \( e \) is an irrational number, natural logarithms are non-repetitive and infinite, similar to \( \pi \).
When working with an exponential function like \( y = a^x \), we can re-express it using the natural logarithm as \( y = e^{x \ln a} \). This conversion relates the original base \( a \) to \( e \), allowing for uniform analysis across different bases.

Conversion Factor

The term "conversion factor" in this context refers to the transformation that adjusts one exponential base to another. Here, the conversion factor is denoted as \( \mu = \ln a \).
In the equation \( y = e^{\mu x} \), this factor converts the exponential function from a base \( a \) to the natural base \( e \). It acts as a scaling factor, perfectly balancing the growth rate change between the bases.
This transformation is essential for simplifying calculations and improving the interpretability of functions in mathematical and real-world applications. A positive conversion factor, \( \ln a > 0 \) when \( a > 1 \), assures that the transformed function retains meaningful and plausible behavior in models such as continuous interest rate calculations or population growth projections.

Base Change

The concept of base change in logarithms allows us to shift an expression from one exponential base to another, often to simplify operations or harmonize with natural processes. When we change the base of an exponential function, we're utilizing logarithmic properties, particularly the change of base formula.
For the function \( y=a^{x} \), the transformation to \( y = e^{\mu x} \) uses the fact that \( a^{x} \) can be re-written as \((e^{\ln a})^{x}\), which simplifies to \( e^{x \ln a} \). This is a specific instance of base change using natural logarithms.
Base transformations are pivotal in exploring complex equations and finding solutions more efficiently. They often assist in deriving integrals and solving differential equations since the natural base \( e \) simplifies many mathematical models, making it more manageable for both theoretical exploration and practical application.