The problem asks to evaluate four expressions using properties of logarithms. The expressions are: 1) $e^{\ln(14) + \ln(10)}$ 2) $10^{\log(14) - \log(2)}$ 3) $11^{\log_{11}(15-12)}$ 4) $12^{13\cdot \log_{12}(2)}$

1. Problem Description

The problem asks to evaluate four expressions using properties of logarithms. The expressions are:

1) $e^{\ln(14) + \ln(10)}$

2) $10^{\log(14) - \log(2)}$

3) $11^{\log_{11}(15-12)}$

4) $12^{13\cdot \log_{12}(2)}$

2. Solution Steps

1) $e^{\ln(14) + \ln(10)}$

Using the logarithm property $\ln(a) + \ln(b) = \ln(ab)$, we have

$e^{\ln(14) + \ln(10)} = e^{\ln(14 \cdot 10)} = e^{\ln(140)}$

Since $e^{\ln(x)} = x$, we have

$e^{\ln(140)} = 140$

2) $10^{\log(14) - \log(2)}$

Using the logarithm property $\log(a) - \log(b) = \log(\frac{a}{b})$, we have

$10^{\log(14) - \log(2)} = 10^{\log(\frac{14}{2})} = 10^{\log(7)}$

Since $10^{\log(x)} = x$, we have

$10^{\log(7)} = 7$

3) $11^{\log_{11}(15-12)}$

First simplify the inside of parenthesis to get $15 - 12 = 3$

$11^{\log_{11}(15-12)} = 11^{\log_{11}(3)}$

Since $a^{\log_a(x)} = x$, we have

$11^{\log_{11}(3)} = 3$

4) $12^{13\cdot \log_{12}(2)}$

Using the logarithm property $a \cdot \log_b(c) = \log_b(c^a)$, we have

$12^{13\cdot \log_{12}(2)} = 12^{\log_{12}(2^{13})}$

Since $a^{\log_a(x)} = x$, we have

$12^{\log_{12}(2^{13})} = 2^{13} = 8192$

3. Final Answer

1) $e^{\ln(14) + \ln(10)} = 140$

2) $10^{\log(14) - \log(2)} = 7$

3) $11^{\log_{11}(15-12)} = 3$

4) $12^{13\cdot \log_{12}(2)} = 8192$