Sure, let's break this down step-by-step using the properties of logarithms.
Given the logarithmic expression:
[tex]\[ \log \left(\frac{x^2 + y}{1000}\right) \][/tex]
We want to use the quotient property of logarithms, which states:
[tex]\[ \log \left(\frac{a}{b}\right) = \log (a) - \log (b) \][/tex]
Here, [tex]\( a \)[/tex] is [tex]\( x^2 + y \)[/tex] and [tex]\( b \)[/tex] is [tex]\( 1000 \)[/tex].
Applying this property to the given expression, we have:
[tex]\[ \log \left(\frac{x^2 + y}{1000}\right) = \log (x^2 + y) - \log (1000) \][/tex]
So, the logarithm of the quotient is the difference of the logarithms of the numerator and the denominator.
Therefore, the final expression is:
[tex]\[ \log (x^2 + y) - \log (1000) \][/tex]
