Question

• Prove that \(\log_2x\approx\ln x+\log_{10}x\) with less than 1% error.

Solution

Start from the right hand side.

$$ \begin{align*} \ln x+\log_{10}x&=\frac{\ln2(\ln x+\log_{10}x)}{\ln2}\\ &=\frac{\ln2(\ln x+\ln x/\ln 10)}{\ln2}\\ &=\left(\ln2+\frac{\ln 2}{\ln 10}\right)\frac{\ln x}{\ln 2}\\ &\approx0.994\log_2x \end{align*} $$