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Chapter 13: Problem 9

Write as the sum or difference of logarithms and simplify, if possible. Assume all variables represent positive real numbers. $$\log _{8}(3 \cdot 10)$$

Short Answer

Expert verified
The given logarithm expression \(\log _{8}(3 \cdot 10)\) can be expressed as the sum of different logarithms and simplified as: \(\log _{8}(3) + \log _{8}(2) + \log _{8}(5)\).

Step by step solution

01

Apply the Product Law of Logarithms

The Product Law states that \(log_b{(mn)} = log_b{m} + log_b{n}\). Using this law, we can separate the expression \(\log _{8}(3 \cdot 10)\) into the sum of two logarithms: \(\log _{8}(3 \cdot 10) = \log _{8}(3) + \log _{8}(10)\) Now, we have the expression as the sum of two logarithms: \(\log _{8}(3) + \log _{8}(10)\).
02

Rewrite the Logarithm

To simplify this expression further, let's rewrite 10 as a product of its prime factors: \(10 = 2 \cdot 5\). Now, we can use the Product Law again to break down the \(\log _{8}(10)\) term: \(\log _{8}(2 \cdot 5) = \log _{8}(2) + \log _{8}(5)\).
03

Combine the Logarithms

Now, let's substitute the simplified form of \(\log _{8}(10)\) back into the original expression: \(\log _{8}(3) + \log _{8}(10) = \log _{8}(3) + (\log _{8}(2) + \log _{8}(5))\). Combine the logarithms into a single expression: \(\log _{8}(3) + \log _{8}(2) + \log _{8}(5)\). This is the simplest form we can express the given logarithm as the sum of different logarithms. The final expression is: \(\log _{8}(3) + \log _{8}(2) + \log _{8}(5)\).

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