Question

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

Answer 1

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 \\\ \end{aligned}$$ Clearly, the relation that we have obtained above is a linear equation in x, so we need to make the coefficient of x equal to 1 to solve this question. Now, dividing both the sides with 4 and cancelling the common factors, we get, **$$\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.