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Chapter 6: Problem 15

Prove that \(\ln (A \cdot B)=\ln A+\ln B,\) where \(A\) and \(B\) are positive real numbers.

Short Answer

Expert verified
The property \[ \text{ln}(A \times B) = \text{ln}(A) + \text{ln}(B) \) is verified by the definition and properties of logarithms and exponentiation.

Step by step solution

01

- Define the Logarithmic Function

Recall the definition of the natural logarithm. For positive real numbers, the natural logarithm function is defined such that \( \text{if } y = \text{e}^x, \text{ then } x = \ln(y) \).
02

- Use Logarithmic Properties

Utilize the property of exponential functions: \( \text{e}^{\text{ln}(A) + \text{ln}(B)} = A \text{ and } B \). According to the properties of exponentiation: \( \text{e}^{\text{ln}(A) + \text{ln}(B)} = A \times B \) and therefore \( \text{ln}(A \times B) = \text{ln}(A) + \text{ln}(B) \).
03

- Confirm the Property

Given that we defined \( C = A \times B \) and used the property \( \text{e}^{\text{ln}(C)} = C \text{ and by substituting one of the results above, yields: \text{e}^{\text{ln}(A) + \text{ln}(B)} = A \times B \). Thus, our property is confirmed by definition and algebra of exponent multiplicator}

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Key Concepts

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

Logarithmic function definition
A logarithmic function is an inverse of an exponential function. For natural logarithms, the base is denoted by Euler's number, approximately 2.71828.
In mathematical terms, the natural logarithm function \( \ln(y) \) is defined such that if \( y = e^x \), then \( x = \ln(y) \).
So, the natural logarithm of a number gives you the power to which you must raise \( e \) to obtain that number.
This definition forms the basis for understanding logarithms in algebra and their properties.
Properties of exponentiation
Exponentiation deals with raising numbers to powers.
Here are important properties of exponentiation:
  • \( e^a \times e^b = e^{a+b} \)
  • \( (e^a)^b = e^{a \times b} \)
  • \( e^0 = 1 \)
  • \( e^{-a} = \frac{1}{e^a} \)
Using these properties, we can simplify complex expressions involving exponents.
Logarithmic identities in algebra
Logarithms have several useful identities that simplify logarithmic expressions:
  • \( \ln(1) = 0 \) because \( e^0 = 1 \)
  • \( \ln(e) = 1 \) because \( e^1 = e \)
  • \( \ln(A \times B) = \ln(A) + \ln(B) \)
  • \( \ln\left(\frac{A}{B}\right) = \ln(A) - \ln(B) \)
  • \( \ln(A^B) = B \ln(A) \)
These identities help in solving logarithmic equations and deriving further algebraic properties.
For example, to prove \( \ln(A \times B) = \ln(A) + \ln(B) \), we rely on the properties of exponentiation and the definition of logarithms.

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Most popular questions from this chapter

Solve for \(t\). a. \(10^{t}=4\) d. \(10^{-t}=5\) b. \(3\left(2^{t}\right)=21\) e. \(5^{t}=7^{t+1}\) c. \(1+5^{t}=3\) f. \(6 \cdot 2^{t}=3^{t-1}\)

Wikipedia is a popular, free online encyclopedia (at \(e n .\) wikipedia.org) that anyone can edit. (So articles should be taken with "a grain of salt.") One Wikipedia article claims that the number of articles posted on Wikipedia has been growing exponentially since October 23,2002 . At that date there were approximately 90 thousand articles posted, and the growth rate was about \(0.2 \%\) per day. a. Create an exponential model for the growth in Wikipedia articles. b. What is the doubling time? Interpret your answer.

For each of the following, find the doubling time, then rewrite each function in the form \(P=P_{0} e^{r t} .\) Assume \(t\) is measured in years. a. \(P=P_{0} 2^{t / 5}\) b. \(P=P_{0} 2^{t / 25}\) c. \(P=P_{0} 2^{2 t}\)

Solve for \(x\). a. \(e^{x}=10\) c. \(2+4^{x}=7\) e. \(\ln (x+1)=3\) b. \(10^{x}=3\) d. \(\ln x=5\) f. \(\ln x-\ln (x+1)=4\)

Expand each logarithm using only the numbers \(2,3, \log 2\), and \(\log 3\). a. \(\log 9\) b. \(\log 18\) c. \(\log 54\)
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