Multiple choice

Solve the system of the equations for $x$ and $y$: $\displaystyle b^{\log x}= a^{\log y}$, where $\displaystyle a> 0,b> 0$ and $\displaystyle a\neq b,ab\neq 1$

  1. $x=\dfrac1a$ and $y=\dfrac1b$
  2. $x=\dfrac1{ab}$ and $y=\dfrac1b$
  3. $x=\dfrac1b$ and $y=\dfrac1a$
  4. $x=\dfrac1{a}$ and $y=\dfrac1{ab}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Take log on both sides: log(x) * log(b) = log(y) * log(a). This is log(x)/log(a) = log(y)/log(b) = k. So x = a^k, y = b^k. If we test x = 1/a, y = 1/b, then log(1/a) * log(b) = -log(a)log(b) and log(1/b) * log(a) = -log(b)log(a). They are equal.

AI explanation

Take the logarithm of both sides of the equation b raised to the power of log x equals a raised to the power of log y to get (log b times log x) equals (log a times log y). Rearranging this yields (log x divided by log a) equals (log y divided by log b), which means log base a of x equals log base b of y. Setting this common value to a constant k results in x equal to a raised to the power of k and y equal to b raised to the power of k, but since the provided options do not contain a parameter k, substituting k equal to negative 1 directly produces x equal to 1 divided by a and y equal to 1 divided by b.

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