Question

For $y = {\log _a}x$ to be defined ‘a’ must be
(a) Any positive real number
(b) Any number
(c) $ \geqslant $ e
(d) Any positive real number $ \ne $ 1

Answer 1

Hint: Here, given is the standard function of logarithm, by the definition of logarithm we can choose the correct option.

Complete step-by-step answer: Given, $y = {\log _a}x$
Logarithmic functions are the inverses of exponential functions. We can write the given function as inverse of the exponential function, as x = ay. The logarithmic function $y = {\log _a}x$ is defined to be equivalent to the exponential equation x = ay. $y = {\log _a}x$ Only under the following conditions: x = ay, a > 0, and a≠1. It is called the logarithmic function with base a.
Also we can understand this situation as, consider the inverse of the exponential function means: x = ay. Given a number x and a base a, to what power y must a be raised to equal x? This unknown exponent, y, equals ${\log _a}x$. So you see a logarithm is nothing more than an exponent.
Any positive real number can’t be the value of a as 1 is also a real number, and logarithm condition is not satisfied by 1. Hence option (a) is incorrect.
Option (b) is incorrect as any real number can’t satisfy the logarithm function.
Option (c) is incorrect as e = 2.732 but value of a can be 2, which is less than e i.e. 2.732.
Therefore, the correct option is (d).

Note: The log of any number less than 1 that has a base of any value will result in a negative number. Remember that the base of the logarithm always has to be positive and greater than 1.