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Chapter 4: Problem 105

Describe the product rule for logarithms and give an example.

Short Answer

Expert verified
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 \(\log_{2}(8 * 4) = \log_{2}(32) = 5\).

Step by step solution

01

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.
02

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.

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Most popular questions from this chapter

Exercises \(86-88\) will help you prepare for the material covered in the first section of the next chapter. $$ \text { Simplify: }-\frac{\pi}{12}+2 \pi $$

Write as a single term that does not contain a logarithm: $$ e^{\ln 8 x^{5}-\ln 2 x^{2}} $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can use any positive number other than 1 in the change-of-base property, but the only practical bases are 10 and \(e\) because my calculator gives logarithms for these two bases.

Graph \(y=\log x, y=\log (10 x),\) and \(y=\log (0.1 x)\) in the same viewing rectangle. Describe the relationship among the three graphs. What logarithmic property accounts for this relationship?

In Exercises \(128-131,\) graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of \(g\) to the graph of \(f\) $$ f(x)=\log x, g(x)=\log (x-2)+1 $$
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