Question

Give an example of a value of $b$ for which $y=b^x$represents exponential decay.

Short answer

The required exponential decay function is$y={\left(\frac{4}{5}\right)}^x$where the value of $b=\frac{4}{5}$.

Step-by-step solution

Step 1. Write down the given information.

The given function is $y=b^x$.

Step 2. Concept used.

The exponential function is of the form $y=a{b}^x$ represents exponential growth if the base $b>1$ and exponential decay if the base $0<b<1$ for all $a>1$.

Step 3. Explanation.

From the concept stated above it can be interpreted that the function $y=b^x$ will represent exponential decay if the value of base (b) of the function $y=b^x$ is such that $0<b<1$. Hence consider $b=\frac{4}{5}$ and plug it in $y=b^x$.

Therefore, $y={\left(\frac{4}{5}\right)}^x$ is the required exponential decay function where the value of $b=\frac{4}{5}$

Step 4. Conclusion.

The required exponential decay function is $y={\left(\frac{4}{5}\right)}^x$ where the value of $b=\frac{4}{5}$.