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How do you solve \[{{\log }_{2}}\left( 4x \right)=5\]?

Last updated date: 29th Feb 2024
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IVSAT 2024
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Hint: Use the basic definition of logarithm given as: - if \[{{\log }_{a}}m=k\] then \[m={{a}^{k}}\]. Using this formula, change the logarithmic expression into the corresponding exponential expression. Evaluate \[{{2}^{5}}\] by multiplying 2 five times and form a linear equation in x. Solve this equation for the value of x to get the answer.

Complete step-by-step solution:
Here, we have been provided with the logarithmic expression: - \[{{\log }_{2}}\left( 4x \right)=5\] and we are asked to solve it. That means we have to find the value of x.
Now, using the basic definition of logarithm mathematically given as: - if \[{{\log }_{a}}m=k\] then \[m={{a}^{k}}\], which can be described as: - if we have ‘a’ as the base of the log and ‘m’ as its argument and the value of log is k then in exponential form the value of the argument is ‘a’ raised to the power ‘k’. So, using this conversion rule, we get,
  & \Rightarrow {{\log }_{2}}\left( 4x \right)=5 \\
 & \Rightarrow 4x={{2}^{5}} \\
 & \Rightarrow 4x=32 \\
Clearly, the relation that we have obtained above is a linear equation in x, so we need to make the coefficient of x equal to 1 to solve this question. Now, dividing both the sides with 4 and cancelling the common factors, we get,
\[\Rightarrow x=8\] Hence, the value of x is 8.

Note: One may note that we can solve the question using a different approach also. We can apply the product to the sum rule of logarithm given as \[\log \left( mn \right)=\log m+\log n\] and simplify the L.H.S. Now, we will substitute the value of \[{{\log }_{2}}4\] equal to 2 and take this numerical value to the R.H.S. to simplify the expression further. The equation then becomes \[{{\log }_{2}}x=3\] and the R.H.S. can be written as \[{{\log }_{2}}8\]. In the final step of the solution we will remove the log function from both the sides and equate the arguments to get the value of x equal to 8.