To rewrite the given expression using the properties of logarithms, we can apply the following logarithmic properties:
1. [tex]\( \log_b(a^c) = c \log_b(a) \)[/tex] (Power Rule)
2. [tex]\( \log_b(a) + \log_b(c) = \log_b(a \cdot c) \)[/tex] (Product Rule)
3. [tex]\( \log_b(a) - \log_b(c) = \log_b(a / c) \)[/tex] (Quotient Rule)
Here is the given expression:
[tex]\[ \log_2(z) + 2 \log_2(x) + 4 \log_9(y) + 12 \log_9(x) - 2 \log_2(y) \][/tex]
### Step-by-Step Transformation:
1. Simplify using the Power Rule:
[tex]\[ 2 \log_2(x) = \log_2(x^2) \][/tex]
[tex]\[ 4 \log_9(y) = \log_9(y^4) \][/tex]
[tex]\[ 12 \log_9(x) = \log_9(x^{12}) \][/tex]
[tex]\[ -2 \log_2(y) = \log_2(1 / y^2) \][/tex]
Substitute these simplifications back into the expression:
[tex]\[ \log_2(z) + \log_2(x^2) + \log_9(y^4) + \log_9(x^{12}) + \log_2(1 / y^2) \][/tex]
2. Combine the logarithms with the same base using the Product and Quotient Rules:
Combine the logarithms with base 2:
[tex]\[ \log_2(z) + \log_2(x^2) + \log_2(1 / y^2) = \log_2(z \cdot x^2 \cdot (1 / y^2)) \][/tex]
Simplify inside the logarithm:
[tex]\[ \log_2((z \cdot x^2) / y^2) \][/tex]
Combine the logarithms with base 9:
[tex]\[ \log_9(y^4) + \log_9(x^{12}) = \log_9(y^4 \cdot x^{12}) \][/tex]
### Final Combined Expressions:
After combining and simplifying, we obtain the following results:
For the logarithms with base 2:
[tex]\[ \log_2((z \cdot x^2) / y^2) \][/tex]
For the logarithms with base 9:
[tex]\[ \log_9(y^4 \cdot x^{12}) \][/tex]
### Summary:
The original expression [tex]\(\log_2(z) + 2 \log_2(x) + 4 \log_9(y) + 12 \log_9(x) - 2 \log_2(y)\)[/tex] simplifies to:
[tex]\[ \log_2((z \cdot x^2) / y^2) \text{ and } \log_9(y^4 \cdot x^{12}) \][/tex]
