Question

How do you solve \[{{\log }_{3}}2+{{\log }_{3}}7={{\log }_{3}}x\]?

Answer 1

Hint: These types of problems can be solved using basic logarithm formulas. First we will simplify the LHS using the formulas we have and then we will apply the formula to get the result from that simplified equation we got.
Let us know some logarithm formulas to solve the question.
\[\log a+\log b=\log \left( a\times b \right)\] when both bases are equal.
If \[\log x=\log y\] then \[x=y\] when both bases are equal.

Complete step by step solution:
From the given question, we are given to solve \[{{\log }_{3}}2+{{\log }_{3}}7={{\log }_{3}}x\].
We can see that on the LHS side we have both the terms with the same base.
So we can apply the above discussed formula here to simplify the LHS.
\[\log a+\log b=\log \left( a\times b \right)\]
We will use this formula here.
By applying the formula we will get
\[\Rightarrow {{\log }_{3}}\left( 2\times 7 \right)={{\log }_{3}}x\]
By multiplying we will get
\[\Rightarrow {{\log }_{3}}14={{\log }_{3}}x\]
Here again we can have two terms equal with the same base.
So we can use the other that is said above to simplify this further.
If \[\log x=\log y\] then \[x=y\].
This is the formula we will use here.
So we have same base as 3 we can write our equation as
\[\Rightarrow 14=x\]
By rewriting it we will get
\[\Rightarrow x=14\]

So by solving the given equation we will get x value as 14.

Note: We must know the formulas to solve this type of questions. Otherwise it would be difficult to figure out the solution. Also we should be careful that we have to apply these formulas only when their bases are equal otherwise we don’t use these formulas.