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Chapter 1: Problem 80

Find the domain of the function \(f(x)=\log _{10}\left(1-\log _{10}\left(x^{2}-5 x+16\right)\right)\)

Short Answer

Expert verified
The domain of the function is [3, 4].

Step by step solution

01

Finding Restrictions from the Innermost Logarithm

The innermost logarithm is \(\log _{10}\left(x^{2}-5 x+16\right)\), which we will call \(g(x)\). From the condition on logarithms, we know that \(g(x)>0\), that is \(x^{2}-5 x+16>0\). To solve this inequality, it is common practice to see if the quadratic can be factorized. In this case, the quadratic factors out as \((x-3)(x-4)>0\). This is a quadratic inequality which divides the number line into three regions (<3, 3-4, >4). Testing a number in each interval shows that the function has positive value only in the interval (3,4).
02

The Outer Logarithm

Once we have the restriction from the innermost logarithm, we can substitute into the outer logarithm: \(\log _{10}\left(1-g(x)\right)\). Since we know that this also must be positive, we can set the inequality \(1- g(x)>0\) that results in \(g(x)<1\). However, from Step 1, we know that \(g(x)>0\) since it is a logarithm expression. Thus, after combining the inequalities, we find \(0< g(x)<1\). Given that \(g(x)=\log _{10}\left(x^{2}-5 x+16\right)\), this results in \(0<\log _{10}(x^{2}-5 x+16)<1\). We already know that \(x\) must be within the interval (3,4) from Step 1. Hence, we test which values in this interval will satisfy this inequality.
03

Solving for the Domain

If the values \(3\) and \(4\) are plugged into \(g(x)\), we find that both boundaries are included in our solution resulting in \(0\leq g(x) \leq 1\). Thus the domain of the original function, \(f(x)=\log _{10}\left(1-\log _{10}\left(x^{2}-5 x+16\right)\right)\), is the closed interval [3, 4].

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