Basic Technical Mathematics, 11th Edition

13.3 Properties of Logarithms

• Sum of Logarithms for Product • Difference of Logarithms for Quotient • Multiple of Logarithm for Power • Logarithms of 1 and b

Because a logarithm is an exponent, it must follow the laws of exponents. The laws used in this section to derive the very useful properties of logarithms are listed here for reference.

$$b^u{b}^v\hspace{0.17em}=\hspace{0.17em}{b}^{u\hspace{0.17em}+\hspace{0.17em}v}$$ (13.4)

$${\displaystyle \frac{b^u}{b^v}}\hspace{0.17em}=\hspace{0.17em}{b}^{u\hspace{0.17em}-\hspace{0.17em}v}$$ (13.5)

$${(b^u)}^n\hspace{0.17em}=\hspace{0.17em}{b}^{nu}$$ (13.6)

A logarithm is an exponent. However, a historical curiosity is that logarithms were developed before exponents were used.

The next example shows the reasoning used in deriving the properties of logarithms.

EXAMPLE 1 Sum of logarithms for product

We know that $8\hspace{0.17em}\times \hspace{0.17em}16\hspace{0.17em}=\hspace{0.17em}128.$ Writing these numbers as powers of 2, we have

$$8\hspace{0.17em}=\hspace{0.17em}{2}^{3}16\hspace{0.17em}=\hspace{0.17em}{2}^{4}128\hspace{0.17em}=\hspace{0.17em}{2}^{}$$