Question

If $x = \log 0.6$ ; $y = \log 1.25$ and $z = \log 3 - 2\log 2$, then the value of ${5^{x + y - z}}$ is
$(a){\text{ 0}}$
$(b){\text{ 2}}$
$(c){\text{ 1}}$
$(d){\text{ 3}}$

Answer 1

Hint: In the above given question, we have to substitute the values given in the expression which is to be evaluated and then by using the various standard identities, proceed ahead to obtain the solution.

Complete step-by-step answer:
We have the given values as

$x = \log 0.6$, $y = \log 1.25$,$z = \log 3 - 2\log 2$.

Now, we have the given expression as

$ = {5^{x + y - z}}$

This can be written as ${5^a}$,

where $a = x + y - z$

Therefore, after substituting the given values in equation (1), we get,

\[a = \log 0.6 + \log 1.25 - (\log 3 - 2\log 2)\]

We know the standard identities

$\log a + \log b = \log (a \times b)$,

$a\log b = {b^a}$

and $\log a - \log b = \log \dfrac{a}{b}$

Thus, by using the above given identities, we get,

 \[ \Rightarrow a = \log (0.6 \times 1.25) - \log \dfrac{3}{4}\]

\[ \Rightarrow a = \log 0.75 - \log 0.75\]

$ \Rightarrow a = 0$

Therefore, by substituting the value of $a$in the given expression, we get

${5^{x + y - z}} = {5^0} = 1$

Hence, the required solution is the option$(c){\text{ 1}}$.

Note: When we face such a type of problem, the key concept is to have an adequate knowledge of the various properties of logarithm. Simply substitute the values in the given expression and with the help of the various logarithmic properties, evaluate the expression to obtain the required solution.