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Chapter 13: Problem 10

Write as the sum or difference of logarithms and simplify, if possible. Assume all variables represent positive real numbers. $$\log _{2}(6 \cdot 5)$$

Short Answer

Expert verified
The simplified expression, using the sum of logarithms, is: \(\log _{2}(6) + \log _{2}(5)\).

Step by step solution

01

Identify the product rule for logarithms

The product rule for logarithms states that the logarithm of a product is the sum of the logarithms of the individual factors. Mathematically, it is written as: $$\log_{a}(xy) = \log_{a}(x) + \log_{a}(y)$$
02

Apply the product rule to the given expression

Using the product rule, rewrite the given expression \(\log _{2}(6 \cdot 5)\) as the sum of individual logarithms: $$\log _{2}(6 \cdot 5) = \log _{2}(6) + \log _{2}(5)$$
03

Simplify, if possible

The resulting expression, \(\log _{2}(6) + \log _{2}(5)\), cannot be simplified further, because neither 6 nor 5 is a power of 2. Therefore, the simplified expression is: $$\log _{2}(6) + \log _{2}(5)$$

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Most popular questions from this chapter

Write as a single logarithm. Assume the variables are defined so that the variable expressions are positive and so that the bases are positive real numbers not equal to \(1 .\) $$\log _{7} 8-4 \log _{7} x-\log _{7} y$$

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