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Chapter 4: Problem 5

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. $$ \log (1000 x) $$

Short Answer

Expert verified
The expanded form of the logarithmic expression \(\log (1000 x)\) is \(3 + \log(x)\).

Step by step solution

01

Identify the properties of logarithms

The product rule of logarithms states that the logarithm of a product is the sum of the logarithms of its factors, i.e., \(\log(ab)=\log(a)+\log(b).\) The power rule states that the logarithm of a number raised to an exponent is the product of the exponent and logarithm of the number, i.e., \(\log(a^n)=n\log(a)\).
02

Apply the product rule

The given expression is \(\log(1000x)\), which can be considered as \(\log(1000)\) and \(\log(x)\) multiplied together. Therefore, by applying the product rule, we can break it into \[\log(1000)+\log(x).\]
03

Apply the power rule and simplify

The number 1000 can be written as \(10^3\). Substituting this into our expression and applying the power rule gives \[\log(10^3)+\log(x) = 3\log(10)+\log(x).\] As the base of our logarithm is \(10\), \(\log(10)\) is equal to \(1\). Therefore, our final expanded expression is \[3+\log(x).\]

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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. $$ \log _{b}\left(\frac{\sqrt[3]{x} y^{4}}{z^{5}}\right) $$

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. $$ 3 \ln x+5 \ln y-6 \ln z $$

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. $$ \log _{7}\left(\frac{7}{x}\right) $$

a. Use a graphing utility (and the change-of-base property) to graph \(y=\log _{3} x\) b. Graph \(\quad y=2+\log _{3} x, \quad y=\log _{3}(x+2), \quad\) and \(y=-\log _{3} x\) in the same viewing rectangle as \(y=\log _{3} x .\) Then describe the change or changes that need to be made to the graph of \(y=\log _{3} x\) to obtain each of these three graphs.

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. $$ \log _{3} 405-\log _{3} 5 $$
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