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Chapter 5: Problem 20

Express as a difference of logarithms. $$\ln \frac{a}{b}$$

Short Answer

Expert verified
\( \text{ln} \frac{a}{b} = \text{ln}(a) - \text{ln}(b) \)

Step by step solution

01

Understand the Logarithm Property

Recall the property of logarithms that states: \ \ \ \ \text{For any positive numbers } a \text{ and } b, \text{ and any base } c, \text{ the logarithm of a quotient is the difference of the logarithms.} \ \ \ \ \[ \text{ln} \frac{a}{b} = \text{ln}(a) - \text{ln}(b) \]
02

Apply the Property

Given \ \ \[ \text{ln} \frac{a}{b} \] \ \ \ \ Apply the logarithm property identified in Step 1: \ \ \[ \text{ln} \frac{a}{b} = \text{ln}(a) - \text{ln}(b) \]

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Key Concepts

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

Natural Logarithm
The natural logarithm, denoted as \(\text{ln}\), is a logarithm with the base 'e', where 'e' is an irrational constant approximately equal to 2.71828. Natural logarithms are commonly used in calculus and complex calculations due to their unique properties.

  • The natural logarithm of a number x is written as \(\text{ln}(x)\)
  • Natural logarithms answer the question: 'To what power must e be raised, to get x?'
For example, \(\text{ln}(e) = 1\) because \(e^1 = e\). It's very useful in continuous growth or decay processes, such as population growths or radioactive decay.
Quotient Rule for Logarithms
The quotient rule for logarithms helps us to compute the logarithm of a quotient of two numbers. The rule states that the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator.

Mathematically, if you have \(a\) divided by \(b\), then the natural logarithm of the quotient is given by:

\[ \text{ln} \frac{a}{b} = \text{ln}(a) - \text{ln}(b) \] This property has crucial applications in simplifying logarithmic expressions. For instance, if \(a = 5\) and \(b = 2\), then: \[ \text{ln} \frac{5}{2} = \text{ln}(5) - \text{ln}(2) \]
Difference of Logarithms
The difference of logarithms is useful for breaking down complex expressions into simpler parts. According to the logarithm properties, when you have a single logarithm representing a quotient, it can be separated into two distinct logarithms: the logarithm of the numerator minus the logarithm of the denominator.

Using the quotient rule, we can express:
\[ \text{ln} \frac{a}{b} = \text{ln}(a) - \text{ln}(b) \]
  • This means we are transforming one logarithm into a more straightforward form.
  • The 'difference of logarithms' technique is particularly beneficial in solving equations or when integrating logarithmic functions. Simplifying expressions this way can often make finding solutions more straightforward.
So, when you see an expression like \( \text{ln} \frac{a}{b} \), remember you can always rewrite it as \[ \text{ln}(a) - \text{ln}(b) \].

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Most popular questions from this chapter

Determine whether each of the following is true or false. Assume that \(a, x, M,\) and \(N\) are positive. $$\log _{a} M-\log _{a} N=\log _{a} \frac{M}{N}$$

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$\ln (x+8)+\ln (x-1)=2 \ln x$$

Salvage Value. \(\quad\) A restaurant purchased a 72 -in. range with six burners for \(\$ 6982 .\) The value of the range each year is \(85 \%\) of the value of the preceding year. After \(t\) years, its value, in dollars, is given by the exponential function \(V(t)=6982(0.85)^{t}\) a) Graph the function. b) Find the value of the range after \(0,1,2,5,\) and 8 years. c) The restaurant decides to replace the range when its value has declined to \(\$ 1000 .\) After how long will the range be replaced?

Using only a graphing calculator, determine whether the functions are inverses of each other. $$f(x)=\sqrt[3]{\frac{x-3.2}{1.4}}, g(x)=1.4 x^{3}+3.2$$

Solve using any method. Given that \(f(x)=e^{x}-e^{-x},\) find \(f^{-1}(x)\) if it exists.
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