Logging the logs

Algebra Level 4

$\large f(x) = \log_4 \log_5 \log_3 (18x-x^2-77)$

What is the integral value of $x$ for which the function above has a finite value?

1 solution

Vishwak Srinivasan
Jun 24, 2015

To shorten out our choices, we will first consider the domain of the first logarithm.

$18x-x^2-77 > 0 \Rightarrow x \in (7,11)$

The integer values in the above set of values of $x$ are 8,9,10 .

On substituting 8 and 10, the first logarithm equals 0, which makes the second logarithm undefined.

On substituting 9, the first logarithm is greater than 1, i.e., $log_34$ .

The second logarithm is also defined and non negative since for the second logarithm to be negative the values should be $\leq 0.2$ .

So 9 is the answer.

It's just as simple as that!

Bonus question : Is there any solution if the function is replaced as $f(x) = \log_6 \log_4 \log_5 \log_3 (18x-x^2-77)$ instead?

In reply to the Moderator, No there is no integral solution.......

In fact, there is no real x which satisfies the given equation.....

Aaghaz Mahajan - 3 years, 2 months ago