To expand the expression [tex]\(\log_a\left(\frac{x^2 y^3}{2^4}\right)\)[/tex] using logarithm properties, we can follow these steps:
1. Use the Property of Logarithms for Division:
We know that the logarithm of a quotient is the difference of the logarithms: [tex]\(\log_a\left(\frac{M}{N}\right) = \log_a M - \log_a N\)[/tex].
Applying this property:
[tex]\[
\log_a\left(\frac{x^2 y^3}{2^4}\right) = \log_a(x^2 y^3) - \log_a(2^4)
\][/tex]
2. Use the Property of Logarithms for Multiplication:
The logarithm of a product is the sum of the logarithms: [tex]\(\log_a(M \cdot N) = \log_a M + \log_a N\)[/tex].
Applying this property:
[tex]\[
\log_a(x^2 y^3) = \log_a x^2 + \log_a y^3
\][/tex]
Thus:
[tex]\[
\log_a\left(\frac{x^2 y^3}{2^4}\right) = \log_a x^2 + \log_a y^3 - \log_a 2^4
\][/tex]
3. Use the Property of Logarithms for Powers:
The logarithm of a power is the exponent times the logarithm of the base: [tex]\(\log_a(M^k) = k \cdot \log_a M\)[/tex].
Applying this property:
[tex]\[
\log_a x^2 = 2 \log_a x
\][/tex]
[tex]\[
\log_a y^3 = 3 \log_a y
\][/tex]
[tex]\[
\log_a 2^4 = 4 \log_a 2
\][/tex]
Therefore:
[tex]\[
\log_a\left(\frac{x^2 y^3}{2^4}\right) = 2 \log_a x + 3 \log_a y - 4 \log_a 2
\][/tex]
Hence, the expanded form of the logarithmic expression is:
[tex]\[
2 \log_a x + 3 \log_a y - 4 \log_a 2
\][/tex]
Looking at the provided options:
- [tex]\(2 \log_a x + 3 \log_a y - 4 \log_a z\)[/tex]
- [tex]\(2 \log_a x - 3 \log_a y + 4 \log_a z\)[/tex]
- [tex]\(\log_a x^2 + \log_a y^3 - 4 \log_a z\)[/tex]
None of these exactly match [tex]\(2 \log_a x + 3 \log_a y - 4 \log_a 2\)[/tex], but the process should be understood step by step as illustrated above.
