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Chapter 5: Problem 100

Explain why \(\log _{a} x\) is defined only for \(01\).

Short Answer

Expert verified
The base of a logarithm significant restricts the numbers it can be. It can't be negative because we cannot consistently get a real output. It can't be 1 because the resulting function is not well-defined for all x. It can't be 0 because logarithm of 0 is undefined. Thus, a can only take values between 0 and 1 (non-inclusive) or greater than 1.

Step by step solution

01

Understand the Real Numbers

The logarithmic function \(\log_{a}(x)\) is defined for all positive real numbers, where x is the number we are taking the log of and 'a' is the base. We are given that a must either be greater than 1 or between 0 and 1, and that x is a real number. This is because the result of the log function must be a real number, and so a and x must be chosen such that the result is real.
02

Understand the Base of a Logarithm

When we say that a is the base of the logarithm, we're essentially saying that a raised to some power equals x. If a is 0, 1 or a negative number, either there would be no possible values of that power (resulting in an undefined answer) or multiple possible values, which goes against the definition of a function which has exactly one output for each input.
03

Reasoning for a

If a was a negative number then when we substitute it into the log function it would not be capable of providing a real number output for all x. This is because negative numbers raised to some powers can result in imaginary numbers. For example, \((-1)^{n/2}\) for any odd integer n gives an imaginary result.
04

Reasoning for a=1

If a was equal to 1, then no matter the power, the result will always be 1, i.e., \(1^{n}=1\) for all real n. In this case, unless x=1, there will be no solution to \(\log_{a}x=n\). Hence the function is not defined for a=1.
05

Reasoning for a=0

If a was equal to 0, then no matter the power, the result will always be 0, i.e., \(0^{n}=0\) for all n not equal to zero. But log of zero is not defined.

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