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Chapter 12: Problem 1

Match the logarithmic equation in Column I with the corresponding exponential equation from Column II. See Example 1. l (a) \(\log _{1 / 3} 3=-1\) (b) \(\log _{5} 1=0\) (c) \(\log _{2} \sqrt{2}=\frac{1}{2}\) (d) \(\log _{10} 1000=3\) (e) \(\log _{8} \sqrt[3]{8}=\frac{1}{3}\) (f) \(\log _{4} 4=1\) ll A. \(8^{1 / 3}=\sqrt[3]{8}\) B. \(\left(\frac{1}{3}\right)^{-1}=3\) C. \(4^{1}=4\) D. \(2^{1 / 2}=\sqrt{2}\) \(\mathbf{E} .5^{0}=1\) F. \(10^{3}=1000\)

Short Answer

Expert verified
(a) with B, (b) with E, (c) with D, (d) with F, (e) with A, (f) with C.

Step by step solution

01

- Understand Logarithmic Form

Recognize that a logarithmic equation \(\text{log}_{b}(a) = c\) means that \(b^c = a\).
02

- Convert Each Logarithmic Equation to Exponential Form

Convert each logarithmic equation into its corresponding exponential form.
03

- Match A: \(\text{log}_{1 / 3} 3=-1\)

Convert \(\text{log}_{1/3}(3) = -1\) using the formula. \(\left(\frac{1}{3}\right)^{-1} = 3\). The matching exponential form is (B).
04

- Match B: \(\text{log}_{5} 1=0\)

Convert \(\text{log}_{5}(1) = 0\) using the formula. \(5^0 = 1\). The matching exponential form is (E).
05

- Match C: \(\text{log}_{2} \sqrt{2}=\frac{1}{2}\)

Convert \(\text{log}_{2}(\sqrt{2}) = \frac{1}{2}\) using the formula. \(2^{\frac{1}{2}} = \sqrt{2}\). The matching exponential form is (D).
06

- Match D: \(\text{log}_{10} 1000=3\)

Convert \(\text{log}_{10}(1000) = 3\) using the formula. \(10^3 = 1000\). The matching exponential form is (F).
07

- Match E: \(\text{log}_{8} \sqrt[3]{8}=\frac{1}{3}\)

Convert \(\text{log}_{8}(\sqrt[3]{8}) = \frac{1}{3}\) using the formula. \(8^{\frac{1}{3}} = \sqrt[3]{8}\). The matching exponential form is (A).
08

- Match F: \(\text{log}_{4} 4=1\)

Convert \(\text{log}_{4}(4) = 1\) using the formula. \(4^1 = 4\). The matching exponential form is (C).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

exponential form
Understanding the exponential form is crucial for solving logarithmic equations. When we have a logarithmic equation like \(\text{log}_{b}(a) = c\), it can be rewritten in exponential form as \[b^c = a\]. This means the base (\(b\)) raised to the power of the result (\(c\)) equals the argument (\(a\)).
For example, if we have \(\text{log}_{2}(8) = 3\), it means that \[2^3 = 8\].
logarithms
Logarithms are the opposite, or inverse, of exponents. They answer the question: to what exponent must we raise the base to obtain a certain number?
In the logarithmic equation, \(\text{log}_{b}(a) = c\), it asks: 'What power of \(b\) gives us \(a\)?' This property is very useful for solving equations where the exponent is unknown.
For example, \(\text{log}_3(27) = 3\), because \({3^3 = 27}\).
logarithmic properties
There are several properties that simplify working with logarithms:
  • Product Property: \(\text{log}_{b}(xy) = \text{log}_{b}(x) + \text{log}_{b}(y)\)
  • Quotient Property: \(\text{log}_{b}(\frac{x}{y}) = \text{log}_{b}(x) - \text{log}_{b}(y)\)
  • Power Property: \(\text{log}_{b}(x^y) = y\text{log}_{b}(x)\)
  • Change of Base Formula: \(\text{log}_{b}(a) = \frac{\text{log}_{k}(a)}{\text{log}_{k}(b)}\)

These properties help simplify complex logarithmic expressions and convert them into more manageable forms.
converting logarithms
To convert a logarithm to its exponential form, remember the basic definition of a logarithm: \[ \text{log}_{b}(a) = c \] and convert it to \[b^c = a\].
Let鈥檚 take \(\text{log}_{10}(100) = 2\) as an example.
This converts to \[10^2 = 100\]. This process helps visualize and understand log equations better.
Similarly, to convert from exponential to logarithmic form, reverse the process. For instance, given \[5^3 = 125\], we write it as \[ \text{log}_{5}(125) = 3 \].

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Most popular questions from this chapter

To solve the equation \(5^{x}=7,\) we must find the exponent to which 5 must be raised in order to obtain \(7 .\) This is \(\log _{5} 7\) (a) Use the change-of-base rule and your calculator to find \(\log _{5} 7\). (b) Raise 5 to the number you found in part (a). What is your result? (c) Using as many decimal places as your calculator gives, write the solution set of \(5^{x}=7\) (Equations of this type will be studied in more detail in Section 12.6.)

If a function is made up of ordered pairs in such a way that the same \(y\)-value appears in a correspondence with two different \(x\)-values, then A. the function is one-to-one B. the function is not one-to-one C. its graph does not pass the vertical line test D. it has an inverse function associated with it.

Let \(k\) represent the number of letters in your last name. (a) Use your calculator to find \(\log k\) (b) Raise 10 to the power indicated by the number in part (a). What is your result? (c) Use the concepts of Section 12.1 to explain why you obtained the answer you found in part (b). Would it matter what number you used for \(k\) to observe the same result?

Find logarithm. Give approximations to four decimal places. \(\log 0.0326\)

The concentration of a drug injected into the bloodstream decreases with time. The intervals of time \(T\) when the drug should be administered are given by $$T=\frac{1}{k} \ln \frac{C_{2}}{C_{1}}$$ where \(k\) is a constant determined by the drug in use, \(C_{2}\) is the concentration at which the drug is harmful, and \(C_{1}\) is the concentration below which the drug is ineffective. (Source: Horelick, Brindell and Sinan Koont, "Applications of Calculus to Medicine: Prescribing Safe and Effective Dosage," UMAP Module 202.) Thus, if \(T=4,\) the drug should be administered every 4 hr. For a certain drug, \(k=\frac{1}{3}, C_{2}=5,\) and \(C_{1}=2 .\) How often should the drug be administered? (Hint: Round down.)
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