Sure, let's break down each expression using properties of logarithms step-by-step:
### Part (a)
Expression: [tex]\( 2 \log_6(2) + \log_6(9) \)[/tex]
1. Apply the power rule of logarithms: The power rule states [tex]\( a \log_b(c) = \log_b(c^a) \)[/tex]. Applying this to [tex]\( 2 \log_6(2) \)[/tex]:
[tex]\[
2 \log_6(2) = \log_6(2^2) = \log_6(4)
\][/tex]
2. Combining the logarithms: Logarithms can be added together using the product rule, which states [tex]\( \log_b(a) + \log_b(c) = \log_b(ac) \)[/tex]. We apply this to our current expression:
[tex]\[
\log_6(4) + \log_6(9) = \log_6(4 \times 9) = \log_6(36)
\][/tex]
3. Evaluate the logarithm: We now have [tex]\( \log_6(36) \)[/tex]. Since [tex]\( 6^2 = 36 \)[/tex], we know:
[tex]\[
\log_6(36) = 2
\][/tex]
Therefore, [tex]\( 2 \log_6(2) + \log_6(9) = 2 \)[/tex].
### Part (b)
Expression: [tex]\( \ln(e^3) - \ln(e^{13}) \)[/tex]
1. Apply the power rule for natural logarithms: The power rule states [tex]\( \ln(a^b) = b \ln(a) \)[/tex]. Applying this to both terms:
[tex]\[
\ln(e^3) = 3 \ln(e) \quad \text{and} \quad \ln(e^{13}) = 13 \ln(e)
\][/tex]
2. Simplify using properties of natural logarithms: Since the natural logarithm of [tex]\( e \)[/tex], [tex]\( \ln(e) \)[/tex], equals 1, we simplify:
[tex]\[
\ln(e) = 1
\][/tex]
So,
[tex]\[
3 \ln(e) = 3 \times 1 = 3 \quad \text{and} \quad 13 \ln(e) = 13 \times 1 = 13
\][/tex]
3. Combine the results: Subtract the two results:
[tex]\[
3 - 13 = -10
\][/tex]
Therefore, [tex]\( \ln(e^3) - \ln(e^{13}) = -10 \)[/tex].
In summary:
- For (a), [tex]\( 2 \log_6(2) + \log_6(9) = 2 \)[/tex]
- For (b), [tex]\( \ln(e^3) - \ln(e^{13}) = -10 \)[/tex]
