Question

question 2
the product property of logarithms works as follows:
\\(\log_b(m \cdot n) = \log_b(m) + \log_b(n)\\)
use this property to rewrite each of the following as a sum of logarithms. do not use the change of base formula. answers should be exact, not decimal approximations.
\\(\ln(gk) = \square\\)
\\(\log_{30}(7wm) = \square\\)
\\(\log(elbow \cdot banana) = \square\\)
note: \\(\log_b(a)\\) can be written using an underscore like this: \\(\log\\_b(a)\\).
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Explanation:

Step1: Apply Product Property to $\ln(gk)$

The natural logarithm $\ln$ has base $e$, so using $\log_b(m \cdot n)=\log_b(m)+\log_b(n)$ with $b = e$, $m = g$, $n = k$:
$\ln(gk)=\ln(g)+\ln(k)$

Step2: Apply Product Property to $\log_{30}(7wm)$

Using the product property with $b = 30$, $m = 7$, $n = wm$ (first split $7wm$ as $7 \cdot w \cdot m$). So:
$\log_{30}(7wm)=\log_{30}(7)+\log_{30}(w)+\log_{30}(m)$

Step3: Apply Product Property to $\log(\text{elbow} \cdot \text{banana})$

Assuming $\log$ is base 10 (common logarithm), using the property with $b = 10$, $m = \text{elbow}$, $n = \text{banana}$:
$\log(\text{elbow} \cdot \text{banana})=\log(\text{elbow})+\log(\text{banana})$

Answer:

$\ln(gk)=\boldsymbol{\ln(g) + \ln(k)}$
$\log_{30}(7wm)=\boldsymbol{\log_{30}(7) + \log_{30}(w) + \log_{30}(m)}$
$\log(\text{elbow} \cdot \text{banana})=\boldsymbol{\log(\text{elbow}) + \log(\text{banana})}$