Let's go through each part of the question step-by-step to evaluate the expressions using properties of logarithms.
### Part (a): [tex]\( 2 \log_6 2 + \log_6 9 \)[/tex]
1. Applying the Power Rule of Logarithms:
The power rule states [tex]\( a \log_b(c) = \log_b(c^a) \)[/tex]. Applying this property:
[tex]\[
2 \log_6 2 = \log_6(2^2)
\][/tex]
2. Simplifying the Expression:
Simplify the exponent:
[tex]\[
\log_6(2^2) = \log_6(4)
\][/tex]
3. Using the Addition Rule of Logarithms:
The addition rule states [tex]\( \log_b(a) + \log_b(c) = \log_b(a \cdot c) \)[/tex]. Applying this property to our expression:
[tex]\[
\log_6(4) + \log_6(9) = \log_6(4 \cdot 9)
\][/tex]
4. Simplifying the Product:
Perform the multiplication inside the logarithm:
[tex]\[
\log_6(4 \cdot 9) = \log_6(36)
\][/tex]
5. Evaluating the Logarithm:
Since [tex]\( 6^2 = 36 \)[/tex], it follows that:
[tex]\[
\log_6(36) = 2
\][/tex]
Therefore, the value of [tex]\( 2 \log_6 2 + \log_6 9 \)[/tex] is [tex]\( 2 \)[/tex].
### Part (b): [tex]\( \ln e^3 - \ln e^{13} \)[/tex]
1. Using the Property of Natural Logarithms:
The natural logarithm of [tex]\( e \)[/tex] raised to a power simplifies as follows: [tex]\( \ln(e^a) = a \)[/tex]. Applying this property to each term:
[tex]\[
\ln(e^3) = 3 \quad \text{and} \quad \ln(e^{13}) = 13
\][/tex]
2. Subtracting the Values:
Substitute the simplified values back into the expression:
[tex]\[
3 - 13
\][/tex]
3. Simplifying the Result:
Perform the subtraction:
[tex]\[
3 - 13 = -10
\][/tex]
Therefore, the value of [tex]\( \ln e^3 - \ln e^{13} \)[/tex] is [tex]\( -10 \)[/tex].
### Final Answers:
(a) [tex]\( 2 \log_6 2 + \log_6 9 = 2 \)[/tex]
(b) [tex]\( \ln e^3 - \ln e^{13} = -10 \)[/tex]
