In Section 8.1 and Section 8.2, we introduced the log operation, which undoes the “raise 10 to the power of” operation. We also saw that we can use the log base \(b\) operation to undo the “raise \(b\) to the power of” operation. This has allowed us to solve exponential equations algebraically.
A logarithmic equation, or a log equation, is an equation that has the variable inside of a log. We will need to solve log equations in the next lesson. In order to solve log equations, we need to know how to undo the log operation. Fortunately, we don’t need another new operation! The “raise \(b\) to the power of” operation undoes the log base \(b\) operation. For our purposes, it will be sufficient to use the “raise 10 to the power of” operation to undo the log (assumed baes 10) operation. That is, we will use the fact
\begin{equation*}
10^{\log(x)} = x
\end{equation*}
to undo the log operation. This fact is true because \(\log(x)\) is the number so that \(10^{\text{that number}} = x\)
Since \(10^{\log(x)} = x\text{,}\) we can rewrite as
\begin{equation*}
x = 10^{4.5}\text{.}
\end{equation*}
Using a calculator to find \(10^{4.5}\text{,}\) we get that \(x \approx 31,622.8\text{.}\) We can check this is correct by using a calculator and finding \(\log(31,622.8) = 4.5000003214\text{.}\) That’s is close enough to \(4.5\) that we can be confident the difference is just from rounding.
We learned how to build a formula to describe a linear trend in Section 4.2. We will need to be able to do this in the next lesson, so let’s practice now.
Every year, the Mauna Loa Observatory in Hawaii measures the levels of carbon dioxide (\(\text{CO}_2\)) in the atmosphere. The graph below shows the annual mean amounts, in parts per million, for the years 2006 - 2022 1
Using your model, predict how much \(\text{CO}_2\) there will be in the atmosphere at Mauna Loa in 2050. How much confidence do you have in your prediction? Explain your answer.
You’ll need to be able to do the following things for this lesson. Rate how confident you are on a scale of 1 - 5 (1 = not confident and 5 = very confident).
