Preview Activity 3.5.1.
In the following questions, we investigate how \(\log_{10}(a \cdot b)\) can be equivalently written in terms of \(\log_{10}(a)\) and \(\log_{10}(b)\text{.}\)
(a)
Write \(10^x \cdot 10^y\) as \(10\) raised to a single power. That is, complete the equation
\begin{equation*}
10^x \cdot 10^y = 10^{\Box}
\end{equation*}
by filling in the box with an appropriate expression involving \(x\) and \(y\text{.}\)
(b)
What is the simplest possible way to write \(\log_{10}10^x\text{?}\) What about the simplest equivalent expression for \(\log_{10}10^y\text{?}\)
(c)
Explain why each of the following three equal signs is valid in the sequence of equalities:
\begin{align*}
\log_{10}(10^x \cdot 10^y) &= \log_{10}(10^{x+y})\\
&= x+y\\
&= \log_{10}(10^x) + \log_{10}(10^y)\text{.}
\end{align*}
(d)
Suppose that \(a\) and \(b\) are positive real numbers so we can think of \(a\) as \(10^x\) for some real number \(x\) and \(b\) as \(10^y\) for some real number \(y\text{.}\) That is, say that \(a = 10^x\) and \(b = 10^y\text{.}\) What does our work in (c) tell us about \(\log_{10}(ab)\text{?}\)
