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.