Question 7 of 10
Write the following expression as a sum and/or difference of logarithms. Express powers as factors.
\[
\log _{c}\left(u^{3} v^{8}\right) \quad u> 0, v> 0
\]
$\log _{c}\left(u^{3} v^{8}\right)=\square($ Simplify your answer. $)$
Answer
\(\boxed{ \log _{c}\left(u^{3} v^{8}\right) = 3\log _{c}(u) + 8\log _{c}(v) }\)
Step 1 ：The given expression is a logarithm of a product. According to the properties of logarithms, the logarithm of a product is equal to the sum of the logarithms of the individual factors.
Step 2 ：Also, the logarithm of a power is equal to the product of the exponent and the logarithm of the base.
Step 3 ：Therefore, we can rewrite the given expression as a sum of logarithms, where the powers are expressed as factors.
Step 4 ：\[ \log _{c}\left(u^{3} v^{8}\right) = 3\log _{c}(u) + 8\log _{c}(v) \]
Step 5 ：\(\boxed{ \log _{c}\left(u^{3} v^{8}\right) = 3\log _{c}(u) + 8\log _{c}(v) }\)
