Question

In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.

${\log }_7\frac{49}{y}$

Short answer

The solution is$2-{\log }_{7}y$.

Step-by-step solution

Step 1. Given and explanation.

We have ${\log }_7\frac{49}{y}$.

The quotient property states that if there is an expression as ${\log }_a\frac{m}{n}$, it is equal to ${\log }_{a}m-{\log }_{a}n$.

Applying this, we get the solution.

Step 2. Finding the solution.

We have ${\log }_7\frac{49}{y}$.

Applying the quotient property, we get
$$\begin{gathered} {\log }_7\frac{49}{y}={\log }_7\left(49\right)-{\log }_7\left(y\right) \\ ={\log }_7{\left(7\right)}^2-{\log }_7\left(y\right) \\ =2-{\log }_7\left(y\right) \end{gathered}$$
Thus we get the value.