The minimum value of the expression log_a(x)+log_x(x) is 2 because the minimum value of each term is 1.
What is a logarithm?
It is another way to represent the power of numbers, and we say that 'b' is the logarithm of 'c' with base 'a' if and only if 'a' to the power 'b' equals 'c'.
[tex]\rm a^b = c\\log_ac =b[/tex]
We have:
1 < a ≤ x
We have an expression:
[tex]\rm log_a(x)+log_x(x)[/tex]
We know,
[tex]\rm log_aa = 1[/tex]
[tex]\rm log_a(x)+1[/tex]
If a = x
Then, [tex]\rm log_a(x)=1[/tex]
We cannot find the function value less than 1 because it goes infinitely in a negative direction.
So the minimum value of the expression:
= 1 + 1
= 2
Thus, the minimum value of the expression log_a(x)+log_x(x) is 2 because the minimum value of each term is 1.
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