Question

How do you solve log 2 + log x = log 3?

Answer 1

We will use the facts of logarithms to combine the both logarithmic functions on the left hand side and then cut off the log from both the sides then to get an equation without logarithmic functions and then solve it.

Complete step-by-step answer:
We are given that we are required to solve the equation given by: log 2 + log x = log 3.
We know that log a + log b = log (a.b).
Replacing a by 2 and b by x to get the following expression:-
$\Rightarrow$log 2 + log x = log 2x
Now, if we put this in the given expression, we will then obtain the following expression:-
$\Rightarrow$log 2x = log 2
Removing the logarithmic functions on both the sides, we will then obtain:-
$\Rightarrow$2x = 2
If we divide both sides by 2, we will then obtain the following expression:-
$\Rightarrow$x = 1

Hence, the required answer is x = 1.

Note:
The students must note that we did not cut off the logarithmic function from both the sides. There is a reason behind the fact that we could take off the logarithmic function from both the sides.
We know that if log a = b, then we can write it as the expression given by: $a = {e^b}$
Now, if we have log x = log y, then we have $x = {e^{\log y}}$
Now, since exponential is raised to some logarithmic power, we can cancel off the logarithmic and exponential to get: x = y and thus we have the required result.
The students must commit to memory the following formulas:-
log a + log b = log (a.b)
If log x = log y, then we have x = y.
If log a = b, then we have $a = {e^b}$.