Properties of Logarithms

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The log of a product is equal to the sum of the individual logs:

\[{\log _b}\left( {MN} \right) = {\log _b}M + {\log _b}N\]

Let us see a justification for this property. Assume that \({\log _b}M = x\) and \({\log _b}N = y\). Thus:

\[\begin{align}&{b^x} = M,\,\,\,{b^y} = N\\& \Rightarrow \,\,\,MN = {b^x} \times {b^y} = {b^{x + y}}\\ &\Rightarrow \,\,\,{\log _b}\left( {MN} \right) = x + y = {\log _b}M + {\log _b}N\end{align}\]

We see that this property is nothing but a manifestation of the fact that when two exponential terms with the same base are multiplied, their powers add. This property is very useful, as it lets us reduce large multiplications to additions. We will soon see how.

Analogous to the product property, we’ve the division property:

\[{\log _b}\left( {\frac{M}{N}} \right) = {\log _b}M - {\log _b}N\]

As you might have anticipated, this property is a result of the fact that when two exponential terms with the same base are divided, their powers subtract. Let us prove this property explicitly now.

Once again, we assume that \({\log _b}M = x\) and \({\log _b}N = y\). Thus:

\[\begin{align}&{b^x} = M,\,\,\,{b^y} = N\\ &\Rightarrow \,\,\,\frac{M}{N} = \frac{{{b^x}}}{{{b^y}}} = {b^{x - y}}\\& \Rightarrow \,\,\,{\log _b}\left( {\frac{M}{N}} \right) = x - y = {\log _b}M - {\log _b}N\end{align}\]

As with the multiplication property, this property lets us convert divisions into subtractions and simplifies calculations.

Another useful property of logarithms concerns exponential terms. This property states that:

\[{\log _b}\left( {{N^e}} \right) = e{\log _b}N\]

That is, the log of \({N^e}\) to any base is e times the log of N to the same base. Let us justify this. Assume that \({\log _b}N = x\) ,We have:

\[\begin{align}&N = {b^x}\,\,\,\; \Rightarrow \,\,\,{N^e} = {\left( {{b^x}} \right)^e} = {b^{e \times x}}\\ &\qquad\qquad\Rightarrow \,\,\,{\log _b}\left( {{N^e}} \right) = e \times x = e{\log _b}N\end{align}\]

This property is a consequence of the fact that when an exponential term is raised to another power, the two powers multiply.

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