Calculus I - Exponential and Logarithm Equations

Show Mobile Notice Show All Notes Hide All Notes

Mobile Notice

You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 1.9 : Exponential And Logarithm Equations

Back to Problem List

9. Find all the solutions to \(\log \left( w \right) + \log \left( {w - 21} \right) = 2\). If there are no solutions clearly explain why.

Show All Steps Hide All Steps

Start Solution

We need to reduce this down to an equation with a single logarithm and to do that we first should rewrite it a little. Upon doing that we can use the basic logarithm properties to combine the two logarithms into a single logarithm as follows,

\[\begin{align*}\log \left( {w\left( {w - 21} \right)} \right) & = 2\\ \log \left( {{w^2} - 21w} \right) & = 2\end{align*}\] Show Step 2

Now all we need to do is exponentiate both sides using 10 (because we’re working with the common logarithm) and then solve for \(y\).

\[\begin{align*}\log \left( {{w^2} - 21w} \right) & = 2\\ {10^{\log \left( {{w^2} - 21w} \right)}} & = {10^2}\\ {w^2} - 21w & = 100\\ {w^2} - 21w - 100 & = 0\\ \left( {w - 25} \right)\left( {w + 4} \right) & = 0\hspace{0.5in} \Rightarrow \hspace{0.5in}w = - 4,\,\,\,\,w = 25\end{align*}\] Show Step 3

We’re dealing with logarithms so we need to make sure that we won’t have any problems with any of our potential solutions. In other words, we need to make sure that if we plug either of the two potential solutions into the original equation we won’t end up taking the logarithm of a negative number or zero.

Upon inspection we can quickly see that if we plug in \(w = - 4\) we will be taking a logarithm of a negative number (in both of the logarithms in this case) and so \(w = - 4\) can’t be a solution. On the other hand, if we plug in \(w = 25\) we won’t be taking logarithms of negative numbers and so \(w = 25\) is a solution.

In summary then, the only solution to the equation is : \(\require{bbox} \bbox[2pt,border:1px solid black]{{w = 25}}\).