Q.Using the laws of logarithms, and given $\log_{10}2=0.3010$ and $\log_{10}3=0.4771$, evaluate $\log_{10}12$ and $\log_{10}1.5$.

Write each number in terms of $2$ and $3$, then apply the product, power and quotient laws of logarithms.

Evaluating $\log_{10}12$. Write $12=2^2\times 3=4\times 3$. Using the product and power laws:

Substituting the given values:

Evaluating $\log_{10}1.5$. Write $1.5=\dfrac{3}{2}$. Using the quotient law:

$\log_{10}1.5=\log_{10}\!\left(\frac{3}{2}\right)=\log_{10}3-\log_{10}2=0.4771-0.3010=0.1761.$ …

A frequent error is writing $\log_{10}12=\log_{10}2+\log_{10}\text{(something)}$ using the wrong factorisation, or mishandling $\log_{10}2^2$ as $(\log_{10}2)^2$ instead of $2\log_{10}2$. …