The provided formula in the image is $y = y_0 e^{kt}$, which is a general form of the exponential growth or decay equation.

The formula $\left(\frac{1}{5}\right)^x$ represents exponential decay. To understand how they got $\frac{1}{5}$, we need to relate it to the standard form of the exponential equation.

Standard Exponential Form

1. Exponential Growth or Decay Formula: $y = y_0 e^{kt}$

   • $y$: Amount of substance at time $t$.
   • $y_0$: Initial amount of substance.
   • $k$: Growth ($k > 0$) or decay ($k < 0$) rate.
   • $t$: Time.

Converting $e^{kt}$ to a Base of $\frac{1}{5}$

The formula $y = y_0 \left(\frac{1}{5}\right)^x$ can be seen as a transformation of the general exponential formula. Here's the step-by-step reasoning:

1. Equivalent Form with Different Base: Exponential decay with base $\frac{1}{5}$ can be written as: $y = y_0 \left(\frac{1}{5}\right)^x$

2. Relating to $e^{kt}$: To convert this into the form $y = y_0 e^{kt}$, we need to express $\left(\frac{1}{5}\right)^x$ in terms of $e$.

3. Logarithmic Conversion: $\left(\frac{1}{5}\right)^x = e^{\ln \left(\left(\frac{1}{5}\right)^x\right)} = e^{x \ln \left(\frac{1}{5}\right)}$

   Here, $\ln \left(\frac{1}{5}\right)$ is the natural logarithm of $\frac{1}{5}$, which is a negative number because $\frac{1}{5} < 1$.

4. Identify $k$: In this expression, $x \ln \left(\frac{1}{5}\right)$ is the exponent. Comparing with the general form $e^{kt}$, we see that: $k = \ln \left(\frac{1}{5}\right)$

   Since $\ln \left(\frac{1}{5}\right)$ is a negative number, it indicates decay.

Summary

The base $\frac{1}{5}$ in the formula $\left(\frac{1}{5}\right)^x$ is derived from the general exponential decay form $y = y_0 e^{kt}$ by using the property of logarithms to convert the exponential base $e$ to the desired base $\frac{1}{5}$.

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Relative Questions

1. How do you convert a general exponential form to a specific base?
2. What is the significance of the decay rate $k$ being negative?
3. How do you determine the half-life of a substance using exponential decay?
4. Can the base of an exponential decay formula be any positive number less than 1?
5. How do you interpret the initial amount $y_0$ in an exponential decay context?
6. What are the applications of exponential decay in real life?
7. How does the value of $k$ affect the rate of decay?
8. How can you graphically represent exponential decay?

Tip

When dealing with exponential functions, remember that changing the base of the exponent will require logarithmic transformations to maintain equivalence. This is useful for simplifying and solving exponential equations.