Answer:
We can start by using logarithmic properties. If a, b, and c are positive numbers, then:
loga^b = b * loga
Using this property, we can rewrite the expression as:
x^(logy - logz) * y^(logz - logx) * z^(logx - logy) = x^(logy/logx - logz/logx) * y^(logz/logy - logx/logy) * z^(logx/logz - logy/logz)
Using the change-of-base formula, we can simplify the exponents:
x^(logy/logx - logz/logx) = (logy/logx) / (logz/logx) = logy/logz
Similarly,
y^(logz/logy - logx/logy) = logz/logx
and
z^(logx/logz - logy/logz) = logx/logy
Now, we can substitute these results back into the original expression:
x^(logy - logz) * y^(logz - logx) * z^(logx - logy) = logy/logz * logz/logx * logx/logy
By the transitive property of logarithms, we have:
logy/logz * logz/logx * logx/logy = logy/logx
Finally, using the change-of-base formula again, we can simplify this result:
logy/logx = y^(1/logx) / x^(1/logx)
Since y and x are positive numbers, their exponents must also be positive. Therefore, we have:
y^(1/logx) / x^(1/logx) = 1
So,
x^(logy - logz) * y^(logz - logx) * z^(logx - logy) = 1
And we have proven that the expression is equal to 1.
Explanation: