Problem 85 - Calculus for Biology and Medicine

Question

Write the following expressions in terms of base \(e\), and simplify: (a) \(3^{x}\) (b) \(4^{x^{2}-1}\) (c) \(2^{-x-1}\) (d) \(3^{-4 x+1}\)

Step-by-Step Solution

Answer

(a) \(e^{x \ln 3}\), (b) \(e^{(x^2 - 1) \ln 4}\), (c) \(e^{-x \ln 2 - \ln 2}\), (d) \(e^{-4x \ln 3 + \ln 3}\).

Step 1: Understanding Exponential Expressions

To express any exponential expression in terms of base \(e\), we need to utilize the property of logarithms where \(a^b = e^{b \ln a}\). This transformation leverages the natural logarithm which enables us to convert the base of an exponential function to \(e\).

Step 2: Convert Expression (a)

For \(3^{x}\), apply the formula: \(3^x = e^{x \ln 3}\). Now, it is expressed in terms of base \(e\).

Step 3: Convert Expression (b)

For \(4^{x^2 - 1}\), apply the conversion to base \(e\): \[4^{x^2 - 1} = e^{(x^2 - 1) \ln 4}\]. This expression is now in terms of base \(e\).

Step 4: Convert Expression (c)

For \(2^{-x-1}\), use the conversion: \[2^{-x-1} = e^{(-x-1) \ln 2} = e^{-x \ln 2 - \ln 2}\]. This shows the expression in terms of base \(e\).

Step 5: Convert Expression (d)

For \(3^{-4x+1}\), apply the formula: \[3^{-4x+1} = e^{(-4x+1) \ln 3} = e^{-4x \ln 3 + \ln 3}\]. Now, the expression is in terms of base \(e\).

Key Concepts

Base e Transformation

When dealing with exponential expressions, converting them to have a base of \(e\) can simplify the processes of calculus and further analysis. This transformation is particularly useful in mathematics because the base \(e\) is related to the natural exponential function, which has properties that simplify many mathematical operations. Here, the transformation utilizes the property of logarithms that allows us to express any power \(a^b\) as \(e^{b \ln a}\). Through this, we translate the base from any other number to \(e\) by using the natural logarithm of the original base:

• Express the exponent in terms of a natural logarithm.
• Apply the property \(a^b = e^{b \ln a}\).

This manipulation is not only handy in simplifying expressions but also transforms them with a consistent base, making subsequent calculations more uniform and manageable.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), where \(e\) is approximately equal to 2.71828. It plays a crucial role in expressing exponential functions with a consistent base. The natural logarithm is the inverse of the `natural exponential function`, which makes it extremely useful in calculus for solving exponential equations and finding anti-derivatives.

• Natural logarithms allow us to transform complex exponentials into linear expressions, making them easier to handle.
• For any number \(a\), \(\ln a\) is the power to which \(e\) must be raised to get \(a\).

This inverse relationship is key in courses that delve into exponential growth or decay, since it simplifies the behavior and analysis of these functions.

Logarithmic Conversion

Logarithmic conversion is a powerful method used in various fields of science and engineering to simplify otherwise complex multiplications and divisions into addition and subtraction. Specifically in mathematics, converting an exponential expression using logarithms like \(\ln\) simplifies expressions, making it possible to transform complex calculations into more manageable forms.

• Begin with identifying the base of the exponential expression.
• Apply the conversion technique such that \(a^b\) becomes \(e^{b \ln a}\).
• This conversion leverages the linear properties of logarithms.

By reducing multiplicative steps through addition during conversion, many mathematical operations that involve exponential expressions are thoroughly simplified.

Exponential Function Simplification

Simplifying exponential functions, particularly those with non-trivial bases, benefits greatly from conversion to a base \(e\). This is due to the inherent simplicity of the natural exponential function's behavior in calculus. To simplify these functions effectively:

• Convert each term to base \(e\) using the formula, resulting in consistent exponential bases.
• A unified base allows more straightforward algebraic manipulation and solving.
• In applied calculus, this leads directly to ease in differentiation and integration, as the derivative of \(e^x\) is simply \(e^x\) itself.

Thus, expressing functions in terms of the natural exponential base not only simplifies them mathematically but also provides a clear path for advanced analytical techniques.