To expand the expression [tex]\(\log_5\left(\frac{25 x^3}{y^7}\right)\)[/tex] using logarithm properties, we can follow these steps:
1. Apply the Quotient Rule:
The logarithm of a quotient is the difference of the logarithms:
[tex]\[
\log_5\left(\frac{25 x^3}{y^7}\right) = \log_5(25 x^3) - \log_5(y^7)
\][/tex]
2. Apply the Product Rule:
The logarithm of a product is the sum of the logarithms:
[tex]\[
\log_5(25 x^3) = \log_5(25) + \log_5(x^3)
\][/tex]
3. Simplify [tex]\(\log_5(25)\)[/tex]:
Recognize that [tex]\(25\)[/tex] is a power of [tex]\(5\)[/tex]:
[tex]\[
25 = 5^2 \quad \text{so} \quad \log_5(25) = \log_5(5^2)
\][/tex]
Using the power rule of logarithms, which states [tex]\(\log_b(a^c) = c \log_b(a)\)[/tex]:
[tex]\[
\log_5(5^2) = 2 \log_5(5)
\][/tex]
Since [tex]\(\log_5(5) = 1\)[/tex] (because any log base [tex]\(b\)[/tex] of [tex]\(b\)[/tex] is 1), we have:
[tex]\[
\log_5(5^2) = 2 \cdot 1 = 2
\][/tex]
4. Simplify [tex]\(\log_5(x^3)\)[/tex]:
Using the power rule again:
[tex]\[
\log_5(x^3) = 3 \log_5(x)
\][/tex]
5. Simplify [tex]\(\log_5(y^7)\)[/tex]:
Again, using the power rule:
[tex]\[
\log_5(y^7) = 7 \log_5(y)
\][/tex]
Putting all these steps together, the expanded form of the given logarithmic expression is:
[tex]\[
\log_5\left(\frac{25 x^3}{y^7}\right) = \log_5(25 x^3) - \log_5(y^7)
\][/tex]
[tex]\[
= (\log_5(25) + \log_5(x^3)) - \log_5(y^7)
\][/tex]
[tex]\[
= (2 + 3 \log_5(x)) - 7 \log_5(y)
\][/tex]
So, the final expanded form is:
[tex]\[
2 + 3 \log_5(x) - 7 \log_5(y)
\][/tex]
