Logarithms in terms of given logarithms

When some logarithms are given as letters, for example \(\lg2=p\) and \(\lg3=q\), any logarithm built from those numbers can be written in terms of the letters.

Method

1. Change the base, if needed, to the base of the given logarithms.
2. Write the number as a product or quotient of powers of the given numbers; the base itself may also appear, since \(\log_a a=1\).
3. Split with the product, quotient and power laws, then replace each given logarithm by its letter.

Example

Given that \(\lg2=p\) and \(\lg3=q\), express (a) \(\lg\sqrt{7.2}\) and (b) \(\log_32\) in terms of \(p\) and \(q\).

\(\begin{aligned} \text{(a)}\quad 7.2&=\dfrac{72}{10}=\dfrac{2^3\times3^2}{10} \\ \lg\sqrt{7.2}&=\tfrac12\left(3\lg2+2\lg3-\lg10\right)=\tfrac12(3p+2q-1) \\ \text{(b)}\quad \log_32&=\dfrac{\lg2}{\lg3}=\dfrac{p}{q} \end{aligned}\)