Question

In the following exercises, use properties of logarithms to write each expression as a sum of logarithms, simplifying if possible

${\log }_{5}25ab$

Short answer

The answer is$=2+{\log }_{5}a+{\log }_{5}b$

Step-by-step solution

Step 1. Given

An expression is given as${\log }_{5}25ab$

Step 2. Calculation

If $M>0,N>0,a>0\&a\ne 1$

${\log }_a(M·N)={\log }_{a}M+{\log }_{a}N$

Use above property,

$$\begin{gathered} {\log }_{5}25ab={\log }_{5}25+{\log }_{5}a+{\log }_{5}b \\ ={\log }_5{5}^2+{\log }_{5}a+{\log }_{5}b \\ =2+{\log }_{5}a+{\log }_{5}b \end{gathered}$$