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Chapter 4: Problem 85

Describe the product rule for logarithms and give an example.

Short Answer

Expert verified
The product rule for logarithms states that \( \log_{b}(mn) = \log_{10}(m) + \log_{10}(n) \). An example is \(\log_{10}(20)\) which equals \( \log_{10}(2) + \log_{10}(10) \), and approximately equals 1.3010.

Step by step solution

01

Definition of the Product Rule for Logarithms

The product rule of logarithms states that the logarithm of a product of numbers is the sum of the logarithms of all those numbers. Mathematically, this rule can be written as \( \log_{b}(mn) = \log_{b}(m) + \log_{b}(n) \) where \( b \) is the base and \( m \) and \( n \) are the numbers.
02

Example illustrating the Product Rule

Let's take an example: suppose we want to find the value of \( \log_{10}(20) \). Using the product rule for logarithms we can break it down as \( \log_{10}(2*10) \). If we apply the product rule now, we get: \( \log_{10}(2) + \log_{10}(10) \)
03

Calculating the individual Logarithms

Now, we can calculate each of these logarithms separately. Since log base 10 of 10 equals 1, our previous step reduce to \( \log_{10}(2) + 1 \). Now, using the value of \( \log_{10}(2) \) as approximately 0.3010, the previous equation becomes \( 0.3010 + 1 = 1.3010 \)

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Most popular questions from this chapter

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log (10,000 x) $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(\frac{x^{2} y}{z^{2}}\right) $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 8 \ln (x+9)-4 \ln x $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \frac{1}{3}\left|2 \ln (x+5)-\ln x-\ln \left(x^{2}-4\right)\right| $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(\frac{V x y^{3}}{z^{3}}\right) $$
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