Question

If $x = \log 0.6$ , $y = \log 1.25$ and $z = \log 3 - 2\log 2$ , then value of $(x + y)$ is equal to
(a) $< 0$
(b) $> 0$
(c) $> 1$
(d) $< 1$

Answer 1

Hint : First we will change the base by using the rule $\log a + \log b = \log a \times b$ . Then we will evaluate all the required terms. Then we will apply the property ${\log _a}a = 1$ . The value of the logarithmic function $\ln e$ is $1$ .

Complete step by step solution:
So, we start by directly applying the property $\log a + \log b = \log a \times b$ .
Hence, we write,
$= x + y \\\ = \log 0.6 + \log 1.25 \\\ = \log (0.6 \times 1.25) \\\ = \log (0.75) \;$
Now if we evaluate the value it comes out as $- 0.1249$ .
Hence, the value of $(x + y)$ is $< 0$ that is option (A).
So, the correct answer is “Option A”.

Note : A logarithm is the power to which a number must be raised in order to get some other number. Example: ${\log _a}b$ here, a is the base and b is the argument. Exponent is a symbol written above and to the right of a mathematical expression to indicate the operation of raising to a power. The symbol of the exponential symbol is $e$ and has the value $2.17828$ . Remember that $\ln a$ and $\log a$ are two different terms. In $\ln a$ the base is e and in $\log a$ the base is $10$ . While rewriting an exponential equation in log form or a log equation in exponential form, it is helpful to remember that the base of exponent.