Problem 54 - Algebra and Trigonometry with Analytic Geometry

Question

Change \(f(x)=100\left(\frac{1}{2}\right)^{x}\) to an exponential function with base \(e\) and approximate the decay rate of \(f\).

Step-by-Step Solution

Answer

The function in base \(e\) is \(f(x) = 100e^{-0.693x}\). The decay rate is approximately \(-0.693\).

Step 1: Identify the components of the equation

The given function is \(f(x) = 100\left(\frac{1}{2}\right)^{x}\). The base of the exponential expression is \(\frac{1}{2}\).

Step 2: Convert base to base e

To convert \(\left(\frac{1}{2}\right)\) to a base \(e\) form, use the identity \(a^x = e^{x\ln a}\). Substitute \(\left(\frac{1}{2}\right)\) for \(a\) and simplify: \(\left(\frac{1}{2}\right)^{x} = e^{x \ln\left(\frac{1}{2}\right)}\).

Step 3: Substitute the conversion back into the function

Now replace \(\left(\frac{1}{2}\right)^{x}\) in the original function: \(f(x) = 100e^{x \ln\left(\frac{1}{2}\right)}\).

Step 4: Calculate the decay rate

Determine \(\ln\left(\frac{1}{2}\right)\): it equals \(-\ln(2)\) since \(\ln(a^b) = b\ln(a)\). Thus, the decay rate is \(-\ln(2)\), an approximate value would be \(-0.693\).

Key Concepts

Base e Conversion

Converting an exponential function to base \(e\) can make it easier to work with, especially if you're dealing with calculus or certain applied math situations. Here's why: the number \(e\) (approximately 2.718) is a universal constant that simplifies various calculations because of its unique mathematical properties.
For the exponential function \(f(x) = 100\left(\frac{1}{2}\right)^{x}\), the base is originally \(\frac{1}{2}\). To convert this to a base of \(e\), we utilize the formula \(a^x = e^{x\ln a}\). This equation tells us that any exponential expression \(a^x\) can be recast in terms of \(e\) using the natural logarithm of the base.

• Substitute \(\frac{1}{2}\) for \(a\).
• Apply the identity to get \(\left(\frac{1}{2}\right)^{x} = e^{x \ln\left(\frac{1}{2}\right)}\).

The new expression within our function \(f(x)\) becomes \(100e^{x \ln\left(\frac{1}{2}\right)}\), completely converting it to base \(e\). This transformation is crucial for further analysis.

Decay Rate

The decay rate in an exponential function specifies how quickly the function decreases over time. A negative decay rate means the function is falling, which is what's happening with the given function.
In our conversion to base \(e\), we found the expression \(e^{x \ln\left(\frac{1}{2}\right)}\). The natural logarithm \(\ln\) of the base \(\left(\frac{1}{2}\right)\) is crucial here. Essentially, \(\ln\left(\frac{1}{2}\right)\) tells us the rate of decay when using base \(e\).

• Compute \(\ln\left(\frac{1}{2}\right)\).
• Realize that \(\ln\left(\frac{1}{2}\right) = -\ln(2)\), which is approximately \(-0.693\).

This negative sign indicates a decay and \(-0.693\) is how quickly it declines per unit increase in \(x\). So, at each step, the function reduces by a rate proportionate to this value.

Natural Logarithm

Natural logarithms are integral to understanding exponential functions, particularly those involving base \(e\). The natural logarithm, noted as \(\ln\), reflects the inverse operation of exponentiation with base \(e\).
In the exercise, \(\ln\left(\frac{1}{2}\right)\) transforms our base from \(\frac{1}{2}\) to \(e\)-based. Essentially:

• \(\ln(x)\) means "logarithm base \(e\) of \(x\)".
• It's the power to which \(e\) must be raised to obtain \(x\).

Using \(\ln\), the transformation process becomes mathematically sound and consistent. Calculating \(\ln\left(\frac{1}{2}\right)\) yields a negative value indicating decay, which aligns with the behavior of the function \(f(x)\). Not only does \(\ln\) simplify conversions, but it also provides a deeper understanding of the function's behavior.

Exponential Decay

Exponential decay describes a process where a quantity decreases at a consistent percentage rate over time. It's frequently occurring in natural settings like radioactive decay or depreciation of asset values.
The function \(f(x) = 100\left(\frac{1}{2}\right)^x\) is a prime example, illustrating a halving mechanism capped by an exponential rate. Once converted to base \(e\), this is represented as \(f(x) = 100e^{x\ln\left(\frac{1}{2}\right)}\).

• The decay rate, \(\ln\left(\frac{1}{2}\right)\), marks the sharpness of this decline.
• The negative logarithmic rate conveys continual proportional reduction.

Understanding this function as an exponential decay helps predict outcomes or values at specific times, which can be crucial in fields like finance, physics, and environmental science. The consistent decay factor ensures a smooth, predictable descent in values.