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If ${{x}^{y}}={{y}^{x}}$ and $x=2y$, then find the value of $x+y$. (Assume that $y\ne 0$)
A. $3$
B. $6$
C. $7$
D. $8$

Last updated date: 15th Jul 2024
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Hint: Take the equation ${{x}^{y}}={{y}^{x}}$, apply $\log $on both sides. After that, simplify it by applying the properties of logarithm and then substitute $x=2y$ in the simplified equation. You will get the value of $y$. Then substitute the value of $y$in $x=2y$, you will get the value of $x$. After getting the values of both $x$ and $y$, put the values in $x+y$. Try it, you will get the answer.

Complete step-by-step answer:
In Mathematics, an equation is a statement that asserts the equality of two expressions. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an equation is defined as containing one or more variables, while in English any equality is an equation.

Solving an equation containing variables consists of determining which values of the variables make the equality true. Variables are also called unknowns and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variable. A conditional equation is only true for particular values of the variables.
Single variable equation can be $x+2=0$.

An equation is written as two expressions, connected by an equals sign ("$=$"). The expressions on the two sides of the equals sign are called the "left-hand side" and "right-hand side" of the equation.

The most common type of equation is an algebraic equation, in which the two sides are algebraic expressions. Each side of an algebraic equation will contain one or more terms. For example, the equation,
has left-hand side $A{{x}^{2}}+Bx+C$ which has three terms, and right-hand side $y$, consisting of just one term. The unknowns are $x$ and $y$ and the parameters are $A,$ $B,$ and $C$.

An equation is analogous to a scale into which weights are placed. When equal weights of something (grain for example) are placed into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, an equal amount of grain must be removed from the other pan to keep the scale in balance. Likewise, to keep an equation in balance, the same operations of addition, subtraction, multiplication and division must be performed on both sides of an equation for it to remain true.

In question we are given the equation ${{x}^{y}}={{y}^{x}}$ and $x=2y$.
Now taking the equation ${{x}^{y}}={{y}^{x}}$ and applying $\log $on both sides we get,
$\log {{x}^{y}}=\log {{y}^{x}}$
We know property of $\log $,
$\log {{a}^{b}}=b\log a$
So applying above property we get,
$y\log x=x\log y$ …… (1)

Now we are given that $x=2y$,

So, substituting $x=2y$ in the equation (1) we get,
$y\log 2y=2y\log y$
Simplifying the equation,
$\log 2y=2\log y$

Now, we know the property that,
$\log ab=\log b+\log a$
So, applying above property we get,
\[\log 2+\log y=2\log y\]
Now, again simplifying the equation we get,
\[\log 2=\log y\]
Now, removing $\log $ from both side we get,
$y=2$ ………………. (2)
Now substituting the value of $y=2$ in $x=2y$ we get,
$x=4$……………… (3)
 So now, we have to find the value of $x+y$,

From (2) and (3), substituting the values of $x$ and $y$ in $x+y$ we get,

Therefore, the value of $x+y$ is $6$.

Read the question carefully. You should be clear with the concept of logarithm. Also, while simplifying don’t jumble yourself. Do not make silly mistakes like missing the terms. Take utmost care that no signs are missed. You should be familiar with all the properties of logarithm.