Question

How do you simplify \[{\log _2}14 - {\log _2}7\]?

Answer 1

Hint: We use the concept and properties of logarithm to solve this question. Since the base is the same on values in the equation, we use the property of subtracting two log values which converts it into division of numbers and one log function. When the base is the same as the value whose log function we have to find then the answer is 1.
* \[\log m - \log n = \log \dfrac{m}{n}\]
* \[{\log _a}a = 1\]

Complete step by step solution:
We have to simplify the equation \[{\log _2}14 - {\log _2}7\] … (1)
We are given two values of log with base 2.
We use the property of logarithm that states that \[\log m - \log n = \log \dfrac{m}{n}\] to solve the equation (1).
\[ \Rightarrow {\log _2}14 - {\log _2}7 = {\log _2}\left( {\dfrac{{14}}{7}} \right)\]
Now we solve the value inside the bracket in right hand side of the equation
We can write \[14 = 2 \times 7\]
\[ \Rightarrow {\log _2}14 - {\log _2}7 = {\log _2}\left( {\dfrac{{2 \times 7}}{7}} \right)\]
Cancel same factors from numerator and denominator in the bracket on right hand side of the equation
\[ \Rightarrow {\log _2}14 - {\log _2}7 = {\log _2}\left( 2 \right)\]
Now we observe that base is same as the numerical value i.e. 2
Since we know the property of logarithm that \[{\log _a}a = 1\]
Then \[{\log _2}2 = 1\]
\[ \Rightarrow {\log _2}14 - {\log _2}7 = 1\]

\[\therefore \] Solution of the equation \[{\log _2}14 - {\log _2}7\] is 1

Note: Many students make the mistake of opening the values using log base on both sides as they try to find the values on the internet or using log tables. Keep in mind we should always apply as many properties of log as we can and in the end when we have an answer as log of a number then look up the value in the log table.
Alternate method:
We have to solve \[{\log _2}14 - {\log _2}7\]
We can also write \[{\log _2}14 = {\log _2}(2 \times 7)\]
Now apply property of log i.e. \[\log m + \log n = \log mn\]
\[ \Rightarrow {\log _2}14 = {\log _2}2 + {\log _2}7\]
Substitute in \[{\log _2}14 - {\log _2}7\]
\[ \Rightarrow {\log _2}2 + {\log _2}7 - {\log _2}7 = {\log _2}2\]
And we know \[{\log _2}2 = 1\]
\[ \Rightarrow {\log _2}14 - {\log _2}7 = 1\]
\[\therefore \] Solution of the equation \[{\log _2}14 - {\log _2}7\] is 1