# Suppose a colony of mushrooms triples in size every 10 days. If there are 10 mushrooms to start,...

## Question:

Suppose a colony of mushrooms triples in size every 10 days. If there are 10 mushrooms to start, how many days until there are 1,000 mushrooms?

## Exponential Growth

A quantity is said to grow exponentially if it increases by a common factor. A general equation for exponential growth is given by {eq}Q=ab^t {/eq} where {eq}a {/eq} is the initial value, {eq}Q {/eq} is the quantity at time {eq}t {/eq} and {eq}b {/eq} is the growth factor. Some examples of exponential growth are populations and compounding interest in investments.

This is an exponential growth problem, since the mushroom population triples every 10 days. We need to write an equation that will multiply our mushroom population by {eq}3 {/eq} every ten days. An exponential equation can be written in the form {eq}Q=ab^t {/eq}. In this problem, our initial value is {eq}a=10 {/eq} and if {eq}t=10 {/eq} we have {eq}Q=30 {/eq}. We can use this information to solve for the growth factor {eq}b {/eq}.

\begin{align*} Q&=ab^t\\ 30&=10b^{10}\\ 3&=b^{10}\\ \sqrt[10]{3}&=b \end{align*}

Now that we have the growth factor, we can write the general model for the mushroom colony growth as {eq}Q=10(\sqrt[10]{3})^t {/eq}. To find how many days until there are 1000 mushrooms, substitute {eq}Q=1000 {/eq}.

\begin{align*} Q&=10(\sqrt[10]{3})^t\\ 1000&=10(\sqrt[10]{3})^t\\ 100&=(\sqrt[10]{3})^t\\ \end{align*}

To solve for a variable that is in the exponent, we need to take the logarithm of both sides of the equation. The type of logarithm does not matter - we will use the common logarithm here.

\begin{align*} 100&=(\sqrt[10]{3})^t\\ \log(100)&=\log((\sqrt[10]{3})^t)\\ \log(100)&=t\log(\sqrt[10]{3})\\ \dfrac{\log(100)}{\log(\sqrt[10]{3})}&=t\\ t&\approx 41.92\\ \end{align*}

Therefore, It will take about 42 days for the mushroom colony to reach 1000 mushrooms.