Question

How do you solve \[\log \left( x+9 \right)-\log \left( x \right)=3\] ?

Answer 1

Hint: These types of logarithmic equations are very simple to solve and very easy to understand. The basic form of logarithm is expressed as:
\[{{\log }_{b}}{{a}^{x}}=x{{\log }_{b}}a\]
Here \[b\] is the base, \[a\] is the exponent and \[x\] is the power.
Now, if \[a=b\] , \[{{\log }_{a}}{{a}^{x}}=x{{\log }_{a}}a=x\] as we know \[{{\log }_{a}}a=1\] .

Complete step by step answer:
We need to keep in mind some of the basic formulae for logarithms. They are:
\[\begin{align}
  & \log a+\log b=\log \left( ab \right) \\
 & \log a-\log b=\log \left( \dfrac{a}{b} \right) \\
\end{align}\]
When no base is given in logarithms, we assume the base to be as \[10\] . We also need to remember that, if \[{{\log }_{10}}a=c\] then we can easily find out the value of \[a\] as, \[a={{10}^{c}}\] . Using all the information stated, we can very easily find out the value of x from the given logarithmic equation. Here in this question we need to apply the second equation as mentioned above.
Applying the second equation in the question, which states,
\[\log a-\log b=\log \left( \dfrac{a}{b} \right)\]
Thus, we can write,
\[\log \left( x+9 \right)-\log \left( x \right)=\log \left( \dfrac{x+9}{x} \right)\]
Now, replacing it with the R.H.S of the above equation, we get,
\[\log \left( \dfrac{x+9}{x} \right)=3\]
\[\begin{align}
  & \Rightarrow \left( \dfrac{x+9}{x} \right)={{10}^{3}} \\
 & \Rightarrow \left( \dfrac{x+9}{x} \right)=1000 \\
 & \Rightarrow x+9=1000x \\
 & \Rightarrow 1000x-x=9 \\
 & \Rightarrow 999x=9 \\
 & \Rightarrow x=\dfrac{9}{999} \\
 & \Rightarrow x=0.009 \\
\end{align}\]
Thus, following these systematic steps, we can very easily find our required solution. Our answer is thus \[x=0.009\] . We can easily cross check our answer by putting this value of \[x=0.009\] in our original given equation.

Note:
We need to keep in mind all the logarithmic formulae for smooth solving of these types of questions. In the solution we have also written that \[\left( \dfrac{x+9}{x} \right)={{10}^{3}}\] , which we can explain as: we can write \[3\] as \[3{{\log }_{10}}10\] . This is because we know \[{{\log }_{10}}10=1\] . After these operations to the question have been done, we can then solve it by the normal way we solve general linear equations.