Question

If $\dfrac{\log a}{(b-c)}=\dfrac{\log b}{(c-a)}=\dfrac{\log c}{(a-b)}$ then prove ${{a}^{a}}\times {{b}^{b}}\times {{c}^{c}}=$

Answer 1

We are given, $\dfrac{\log a}{(b-c)}=\dfrac{\log b}{(c-a)}=\dfrac{\log c}{(a-b)}$. Assume $\dfrac{\log a}{(b-c)}=\dfrac{\log b}{(c-a)}=\dfrac{\log c}{(a-b)}=k$, where $k$ is constant. So, we get, $\log a=k(b-c)$, $\log b=k(c-a)$ and $\log c=k(a-b)$. Taking ${{a}^{a}}\times {{b}^{b}}\times {{c}^{c}}$ and apply log on the expression and simplify it.

Complete step-by-step answer:
We are given, $\dfrac{\log a}{(b-c)}=\dfrac{\log b}{(c-a)}=\dfrac{\log c}{(a-b)}$.
Let us assume $\dfrac{\log a}{(b-c)}=\dfrac{\log b}{(c-a)}=\dfrac{\log c}{(a-b)}=k$, where $k$ is constant.
Now we get,
$\log a=k(b-c)$ …………. (1)
$\log b=k(c-a)$ ………… (2)
$\log c=k(a-b)$ ………… (3)
Now, taking ${{a}^{a}}\times {{b}^{b}}\times {{c}^{c}}$ and applying log to the expression we get,
$\log ({{a}^{a}}\times {{b}^{b}}\times {{c}^{c}})=\log ({{a}^{a}})+\log ({{b}^{b}})+\log ({{c}^{c}})$
Now we know that, $\log {{x}^{y}}=y\log x$, now applying the property we get,
$\log ({{a}^{a}}\times {{b}^{b}}\times {{c}^{c}})=a\log (a)+b\log (b)+c\log (c)$
Now applying (1), (2) and (3) in above,
$\log ({{a}^{a}}\times {{b}^{b}}\times {{c}^{c}})=ak(b-c)+bk(c-a)+ck(a-b)$
Now simplifying we get,
$\log ({{a}^{a}}\times {{b}^{b}}\times {{c}^{c}})=akb-akc+bkc-bka+cka-ckb$
$\log ({{a}^{a}}\times {{b}^{b}}\times {{c}^{c}})=0$
Now taking antilog we get,
$({{a}^{a}}\times {{b}^{b}}\times {{c}^{c}})={{e}^{0}}$
Also, we know, ${{a}^{0}}=1$,
${{a}^{a}}\times {{b}^{b}}\times {{c}^{c}}=1$
If $\dfrac{\log a}{(b-c)}=\dfrac{\log b}{(c-a)}=\dfrac{\log c}{(a-b)}$ then ${{a}^{a}}\times {{b}^{b}}\times {{c}^{c}}=1$.

Additional information:
A logarithm is defined as the power to which number must be raised to get some other values. It is the most convenient way to express large numbers. A logarithm has various important properties that prove multiplication and division of logarithms can also be written in the form of logarithm of addition and subtraction. Logarithms are nowadays widely used in the field of science and technology. We can even find logarithmic calculators which have made our calculations much easier. These find its applications in surveying and celestial navigation purposes. The logarithm of any positive number, whose base is a number, which is greater than zero and not equal to one, is the index or the power to which the base must be raised in order to obtain the given number.

Note: The properties of logarithm should be known. Here are some properties of logarithm,
$\log (xy)=\log x+\log y$
$\log {{x}^{y}}=y\log x$.