Question

Which of the following real numbers is (are) non-positive?
A) ${\log _{0.3}}\left( {\dfrac{{\sqrt 5 + 2}}{{\sqrt 5 - 2}}} \right)$
B) ${\log _7}\left( {\sqrt {83} - 9} \right)$
C) ${\log _7}\left( {\cot \dfrac{\pi }{8}} \right)$
D) ${\log _2}\sqrt {9.\sqrt[3]{{{{27}^{\dfrac{{ - 5}}{3}}}{{.243}^{\dfrac{{ - 7}}{5}}}}}}$

Answer 1

by the condition, any positive real number as a base to be raised to any real power, always producing a positive result, so ${\log _b}(x)$ for any two positive real numbers b and x, where b is greater than 1, is always a unique real number y, we need to check the given numbers.

Complete step by step solution: The logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x
More generally, exponentiation allows any positive real number as a base to be raised to any real power, always producing a positive result, so${\log _b}(x)$ for any two positive real numbers b and x, where b is greater than 1, is always a unique real number y.
So now we can check the above conditions for given options.

In option a
$\Rightarrow {\log _{0.3}}\left( {\dfrac{{\sqrt 5 + 2}}{{\sqrt 5 - 2}}} \right)$
Here our b = 0.3 and $x = \dfrac{{\sqrt 5 + 2}}{{\sqrt 5 - 2}}$
So here our base is less than 1
It gives us a non positive value
And in our other options, the base is greater than 1.
Hence they are positive.

The correct option is (a).

Note: The logarithm base 10 (that is b = 10) is called the common logarithm and is commonly used in science and engineering. The natural logarithm has the number e as its base; its use is widespread in mathematics and physics, because of its simpler integral and derivative.