Problem 105
Question
Describe the product rule for logarithms and give an example.
Step-by-Step Solution
Verified Answer
The product rule for logarithms states that the logarithm of a product is equal to the sum of the logarithms of its factors. An example is \(\log_{2}(8 * 4) = \log_{2}(32) = 5\).
1Step 1: Product Rule for Logarithms
The product rule for logarithms states that the logarithm of the product of two numbers is equal to the sum of the logarithms of the individual numbers. Algebraically, this can be written as: \(\log_b (xy) = \log_b(x) + \log_b(y)\), where b is the base, x and y are the arguments of the logarithms.
2Step 2: Example of the Product Rule for Logarithms
Let's consider an example. If we have \(\log_{2}(8) * \log_{2}(4)\), using the product rule for logarithms, we can rewrite this as \(\log_{2}(8*4)\), which simplifies to \(\log_{2}(32)\). The log base 2 of 32 equals 5, so this becomes five.
Other exercises in this chapter
Problem 104
The formula \(A=25.1 e^{0.0187 t}\) models the population of Texas, \(A,\) in millions, \(t\) years after 2010 . a. What was the population of Texas in \(2010 ?
View solution Problem 104
In Exercises 101–104, write each equation in its equivalent exponential form. Then solve for x. $$ \log _{64} x=\frac{2}{3} $$
View solution Problem 105
In Exercises 105–108, evaluate each expression without using a calculator. $$ \log _{3}\left(\log _{7} 7\right) $$
View solution Problem 106
Describe the quotient rule for logarithms and give an example.
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