Sure, let's go through the steps to rewrite the expression [tex]\(\log \left(\frac{x^3 y^{19}}{z^{19}}\right)\)[/tex] using the Laws of Logarithms:
1. Quotient Rule: The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.
[tex]\[
\log \left(\frac{x^3 y^{19}}{z^{19}}\right) = \log(x^3 y^{19}) - \log(z^{19})
\][/tex]
2. Product Rule: The logarithm of a product is the sum of the logarithms of the factors.
[tex]\[
\log(x^3 y^{19}) = \log(x^3) + \log(y^{19})
\][/tex]
3. Power Rule: The logarithm of a number raised to a power is the power times the logarithm of the number.
[tex]\[
\log(x^3) = 3 \log(x)
\][/tex]
[tex]\[
\log(y^{19}) = 19 \log(y)
\][/tex]
[tex]\[
\log(z^{19}) = 19 \log(z)
\][/tex]
4. Combine the terms: Substitute back into the expression:
[tex]\[
\log \left(\frac{x^3 y^{19}}{z^{19}}\right) = \log(x^3 y^{19}) - \log(z^{19})
\][/tex]
[tex]\[
\log(x^3 y^{19}) = 3 \log(x) + 19 \log(y)
\][/tex]
Therefore,
[tex]\[
\log \left(\frac{x^3 y^{19}}{z^{19}}\right) = 3 \log(x) + 19 \log(y) - 19 \log(z)
\][/tex]
So we have:
[tex]\[
\log \left(\frac{x^3 y^{19}}{z^{19}}\right) = 3 \log(x) + 19 \log(y) - 19 \log(z)
\][/tex]
In this form,
- [tex]\(A = 3\)[/tex]
- [tex]\(B = 19\)[/tex]
- [tex]\(C = -19\)[/tex]
Thus, the expression [tex]\(\log \left(\frac{x^3 y^{19}}{z^{19}}\right)\)[/tex] is rewritten as [tex]\(A \log (x) + B \log (y) + C \log (z)\)[/tex] with [tex]\(A = 3\)[/tex], [tex]\(B = 19\)[/tex], and [tex]\(C = -19\)[/tex].
