Question

How do you solve $ \log \left( {x + 10} \right) - \log \left( x \right) - 2\log \left( 5 \right)? $

Answer 1

Hint: In this question we have to find the $ x $ value from the above logarithmic equation. For that we are going to simplify the equation. Next, we use some laws of logarithms and then simplify to arrive at our final answer. Next, we rearrange the variables and numbers. Finally we get the required answer.

Formula used: Quotient rule:
The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.
  $ {\log _a}\dfrac{A}{B} = {\log _a}A + {\log _a}B $ .
Power rule:
 $ {\log _a}{A^n} = n{\log _a}A $

Complete step-by-step solution:
It is given that $ \log \left( {x + 10} \right) - \log \left( x \right) - 2\log \left( 5 \right) $
To find the $ x $ value:
Our first step is to rewrite the equation using some laws of logarithms, specifically, $ \log A - \log B = \log \left( {\dfrac{A}{B}} \right). $
This law allows us to rewrite the left-hand side of the above equation as
 $ \Rightarrow \log \left( {\dfrac{{x + 10}}{x}} \right) = 2\log 5. $
Next, we use another law of logarithms, $ A\log B = \log {B^A} $ , allows us to rewrite the right-hand side equivalently as
 $ \Rightarrow \log \left( {\dfrac{{x + 10}}{x}} \right) = \log {5^2} $
Now, squaring the number in right hand side and we get
 $ \Rightarrow \log \left( {\dfrac{{x + 10}}{x}} \right) = \log 25 $
Now, you did not specify a base for the log function here, so I will assume that log means base-2 logarithm. Still, whether the base is 2, 10, e or whatever, it actually does not matter. The answer will be the same.
Therefore, we raise 2 to both sides in the above equation and we get
 $ \Rightarrow {2^{\log \left( {\dfrac{{x + 10}}{x}} \right)}} = {2^{\log \left( {25} \right)}} $
The exponential and the logarithm are inverse functions, so the base-2 and the logarithms will cancel:
 $ \Rightarrow \left( {\dfrac{{x + 10}}{x}} \right) = 25 $
From here, we just need to use some simple algebra, multiplying both sides by $ x $ :
 $ \Rightarrow x + 10 = 25x $
Next, rearranging the $ x $ terms and then subtracting $ x $ from both sides and we get
 $ \Rightarrow 10 = 25x - x $
On subtracting we get
 $ \Rightarrow 10 = 24x $
Next, we move 24 into the right hand side and we get
 $ \Rightarrow \dfrac{{10}}{{24}} = x $
And then simplify to arrive at our final answer $ x $ :
 $ \Rightarrow x = \dfrac{5}{{12}} $

Therefore, the final answer is $ x = \dfrac{5}{{12}}. $

Note: In this question, we will learn common logarithms and how to solve problems using common logarithms. It means that logarithms to base 10 are called common logarithms.
We often write Common logarithms can be evaluated using a scientific calculator. Recall that by the definition of logarithm.
 $ \log Y = X \Leftrightarrow Y = {10^X} $