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Chapter 3: Problem 109

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

Short Answer

Expert verified
The base 'a' of an exponential function (and hence of a logarithmic function) cannot be negative, 0, or 1. 'a' should be either between 0 and 1 or greater than 1 because a negative, zero or 1 value for the base can't accommodate every possible positive number for x, according to the definition of logarithms.

Step by step solution

01

Recalling the definition of logarithms

According to Berkeley's definition of logarithms, if \(y = log_b x\), then \(x = b^y\). This is important because the constraints on the base 'a' directly influence the values that x can take.
02

Limitations of base a

Firstly, the base of a logarithm cannot be negative nor 0 because neither negative numbers nor zero can be raised to any power to produce a real number. So, the base a is strictly positive.
03

Base of a logarithm can't be 1

A base of 1 also fails to work because 1 raised to any power is always 1. The values of x would therefore have to be 1, no matter what y is, obviously contradicting the definition of a function where each input corresponds to exactly one output.
04

Why base a has to be greater than 0 and not 1

If 01, each time the exponent increases by 1, the result gets larger. This makes it possible to accept any positive number as x and have a corresponding y value.

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Most popular questions from this chapter

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