Answer:
6
Step-by-step explanation:
The given function to us is ,
[tex]\rm\implies f(x)= log_a(x) [/tex]
And its value at 7 is 2 , that is ,
[tex]\rm\implies f(x)= log_a(7) =2[/tex]
Taking this ,
[tex]\rm\implies 2= log_a(7) [/tex]
In general we know that ,
[tex]\bf\to log_a b = c ,\ then \ a^c = b [/tex]
Using this , we have ,
[tex]\rm\implies a^2 = 7 [/tex]
Squarerooting both sides ,
[tex]\rm\implies a =\sqrt{ 7 }[/tex]
Therefore , when x is 343 ,
[tex]\rm\implies f(343)= log_{\sqrt7} ( 343) [/tex]
We can write , 343 as 7³ ,
[tex]\rm\implies f(343)= log_{\sqrt7}7^3 [/tex]
[tex]\rm\implies f(343)= log_{7^{\frac{1}{2}}} 7^3 [/tex]
This can be written as ,
[tex]\rm\implies f(343)= \dfrac{ 3}{\frac{1}{2}} [/tex]
[tex]\rm\implies \boxed{\blue{\rm f(343)= 6 }}[/tex]
Hence the required answer is 6.
