Question

If $x = \log 0.6$, $y = \log 1.25$ and $z = \log 3 - 2\log 2$, then the value of $\left( {x + y} \right)$ is equal to:
A) $ < 0$
B) $ > 0$
C) $ > 1$
D) $ < 1$

Answer 1

Hint:
We have given different values of different variables in terms of the logarithms. We will use the given values and interpret the given result. We will use basic properties of operations on logarithms to derive the required values in terms of logarithms. Then we need to observe the behaviour of the function in order to obtain the result in terms of required options.

Complete step by step solution:
First, we will start by writing the given data.
We have given values for three variables.
The given values are $x = \log 0.6$, $y = \log 1.25$ and $z = \log 3 - 2\log 2$.
We have to interpret the value of the addition $\left( {x + y} \right)$.
We will use the property of logarithm which relates the addition and multiplication of the logarithms.
For any two numbers $a$ and $b$ the following relation holds:
$\log a + \log b = \log ab$
Given that all the values are well defined.
We will start with writing the given relation:
$x + y = \log 0.6 + \log 1.25$
Using the above-mentioned property, we can write the following:
$x + y = \log \left( {0.6 \times 1.25} \right)$
On simplifying we write:
$x + y = \log 0.75$
We know that $\log 1 = 0$ and $0.75 < 1$ therefore, $\log 0.75 < 0$.
Observe that the value is obviously going to be less than one as well but we have to select the most correct answer out of the given choice.

Hence, the correct option is A.

Note:
Note that the example is very trivial but the answer is a little bit tricky as the options are not in the standard form. Also use the properties of logarithms very carefully. You can observe that we don’t have to take antilogarithm any point in order to reach till the final answer.