Question

How do you solve ${{\log }_{2}}\left( x+2 \right)-{{\log }_{2}}\left( x-5 \right)=3$?

Answer 1

Hint: We will use the properties of logarithm to solve this question and find the value of unknown i.e. x. We will use the quotient rule of the logarithm which is given as
Quotient rule – $\log \left( \dfrac{m}{n} \right)=\log m-\log n$

Complete step by step answer:
We have been given an expression ${{\log }_{2}}\left( x+2 \right)-{{\log }_{2}}\left( x-5 \right)=3$.
We have to solve the given expression and find the value of x.
Now, we know that if base of logarithm is same then by quotient rule $\log \left( \dfrac{m}{n} \right)=\log m-\log n$
Now, applying the quotient rule to the LHS of the given expression we will get
\[\begin{align}
  & \Rightarrow {{\log }_{2}}\left( x+2 \right)-{{\log }_{2}}\left( x-5 \right)=3 \\
 & \Rightarrow {{\log }_{2}}\dfrac{\left( x+2 \right)}{\left( x-5 \right)}=3 \\
\end{align}\]
Now, we know by property of logarithm that
$\begin{align}
  & {{\log }_{a}}x=y \\
 & \Rightarrow {{a}^{y}}=x \\
\end{align}$
Now, applying the property to the obtained equation we will get
\[\Rightarrow {{2}^{3}}=\dfrac{\left( x+2 \right)}{\left( x-5 \right)}\]
Now, simplifying the above obtained equation we will get
\[\Rightarrow 8=\dfrac{\left( x+2 \right)}{\left( x-5 \right)}\]
Now, cross multiplying the above obtained equation we get
\[\Rightarrow 8\left( x-5 \right)=\left( x+2 \right)\]
Now, let us open the parenthesis on the LHS and then simplifying further we will get
\[\Rightarrow 8x-40=\left( x+2 \right)\]
Now, on solving the algebraic operations we will get
\[\begin{align}
  & \Rightarrow 8x-40=x+2 \\
 & \Rightarrow 8x-x=2+40 \\
 & \Rightarrow 7x=42 \\
 & \Rightarrow x=\dfrac{42}{7} \\
 & \Rightarrow x=6 \\
\end{align}\]

So, on solving the given expression ${{\log }_{2}}\left( x+2 \right)-{{\log }_{2}}\left( x-5 \right)=3$ we get the value of variable $x=6$.

Note: We know that logarithm is the inverse function to the exponentiation. We will use the basic properties of logarithm to solve further. If the base of logarithm is not given in the question we assume that it is a common logarithm. Logarithm of base 10 is called the common logarithm. Logarithms of base e are called natural logarithms. Avoid basic calculation mistakes.