$ln\left(\frac{x+y}{2}\right) = \frac{1}{2} [ln\left(x\right) + ln\left(y\right)]$
then, the value of $\frac{x}{y} + \frac{y}{x}$ is
Logarithmic Equation Simplification
We are given the equation for positive non-zero real variables $x$ and $y$: $ ln\left(\frac{x+y}{2}\right) = \frac{1}{2} [ln\left(x\right) + ln\left(y\right)] $
Applying Logarithm Properties
Using the logarithm property $ln(a) + ln(b) = ln(ab)$, the right side becomes:
$ \frac{1}{2} [ln\left(x\right) + ln\left(y\right)] = \frac{1}{2} ln(xy) $Using the property $c \cdot ln(a) = ln(a^c)$, this simplifies to:
$ ln((xy)^{\frac{1}{2}}) = ln(\sqrt{xy}) $Equating Arguments
The equation now is: $ ln\left(\frac{x+y}{2}\right) = ln(\sqrt{xy}) $ Since the natural logarithm function is one-to-one, we can equate the arguments:
$ \frac{x+y}{2} = \sqrt{xy} $Solving for Variable Relationship
Multiply both sides by 2:
$ x+y = 2\sqrt{xy} $Square both sides to eliminate the square root:
$ (x+y)^2 = (2\sqrt{xy})^2 $ $ x^2 + 2xy + y^2 = 4xy $Rearrange the terms to form a quadratic expression:
$ x^2 - 2xy + y^2 = 0 $Factor the expression, which is a perfect square:
$ (x-y)^2 = 0 $Taking the square root gives:
$ x-y = 0 $ $ x = y $Calculating the Required Value
We need to find the value of $\frac{x}{y} + \frac{y}{x}$. Since we found $x=y$, we can substitute $y$ with $x$ (or vice versa):
$ \frac{x}{x} + \frac{x}{x} = 1 + 1 $ $ = 2 $Therefore, the value of $\frac{x}{y} + \frac{y}{x}$ is 2.
Important Questions from Algebra
- If $Pe^x = Qe^{-x}$ for all real values of $x$, which one of the following statements is true?
The relationship between two variables $x$ and $y$ is given by $x + py + q = 0$ and is shown in the figure. Find the values of $p$ and $q$.
Note: The figure shown is representative.
The real variables $x, y, z$ and the real constants $p, q, r $ satisfy
$\frac{x}{pq - r^2} = \frac{y}{qr - p^2} = \frac{z}{rp - q^2}$
Given the denominators are non-zero, the value of $px + qy + rz$ isThe complex function
$e^{-\left(\frac{2}{z-1}\right)}$
has __________________Consider two matrices: $P = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ and $Q = \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}$.
Which of the following statement is/are true?
