/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 74 \(\log _{2 / 3} 6\)... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 74

\(\log _{2 / 3} 6\)

Short Answer

Expert verified
The result of \(\log _{2 / 3} 6\) is approximately -1.63093. It is a negative number because the base \(2/3\) is less than 1 and we are finding the exponential value that gives a number larger than the base.

Step by step solution

01

Understand the logarithm operation

First you need to understand what the given logarithmic operation represents. It represents the power or exponent to which the base \(2/3\) must be raised to give 6.
02

Using the properties of logarithm

The logarithmic expression \(\log _{2 / 3} 6\) can be rewritten using the change of base formula. According to the change of base formula, the logarithm base \(b\) of \(a\) is equal to the natural log of \(a\) divided by the natural log of \(b\). So, in our case, it becomes \(\log _{2 / 3} 6 = \frac{\ln 6}{\ln 2/3}\).
03

Calculate using a natural logarithm

Now you can use the natural logarithms to solve the expression. The natural log of 6 divided by the natural log of \(2/3\). This gives the numerical value of the expression.

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Key Concepts

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

Change of Base Formula
The change of base formula is a convenient method for evaluating logarithms that are not accessible directly on a calculator. The formula converts a logarithm of any base to a fraction of logarithms using a different base, often the natural logarithm base.
  • The basic formula is: \[\log_b a = \frac{\log_c a}{\log_c b}\]where \(b\) is the original base, \(a\) is the value of the logarithm, and \(c\) is the new base you鈥檙e changing to. Typically, we use \(c = 10\) (common logarithm) or \(c = e\) (natural logarithm).
  • This transformation makes it possible to use a calculator to find the values of logarithms like \(\log_{2/3} 6\) by evaluating \(\ln 6 / \ln (2/3)\).
Changing the base simplifies solving logarithms in calculators, especially when the base is a fraction or an uncommon number.
Natural Logarithm
The natural logarithm is a special logarithm that uses the base \(e\), an irrational constant approximately equal to 2.71828. It is denoted by \(\ln\), such that \(\ln x = \log_e x\).
  • This form of a logarithm is heavily used in calculus and complex mathematics, appearing in formulas involving growth rates and decay.
  • The natural logarithm simplifies computational processes and equations involving exponential functions of \(e\), making it essential for scientific calculations.
  • When using natural logarithms in the change of base formula, as in solving \(\log_{2/3} 6 = \frac{\ln 6}{\ln (2/3)}\), it allows you to use foundational mathematical principles to evaluate complex expressions more seamlessly.
Natural logarithms are critical for efficiently handling calculations involving growth patterns and natural processes.
Properties of Logarithms
Logarithms have several properties that simplify mathematical operations. Understanding these properties can make solving logarithmic problems easier.
  • Product Rule: \(\log_b(xy) = \log_b x + \log_b y\).
  • Quotient Rule: \(\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y\).
  • Power Rule: \(\log_b(x^k) = k \cdot \log_b x\).
  • Change of Base Formula: Useful for converting complex logarithms to a base that is easier to work with, such as from \(\log_{2/3} 6\) to \(\frac{\ln 6}{\ln (2/3)}\).
These properties and formulas serve as fundamental tools for manipulating expressions and performing calculations in algebra, facilitating better comprehension and problem-solving skills.

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Most popular questions from this chapter

\(r=3+6 \sin \theta\)

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{ll} {\text { Conic }} & \text { Vertex or Vertices } \\ \text { Parabola } & (10, \pi / 2) \\ \end{array} $$

\(r=2 \sec \theta\)

\(r=4(1+\sin \theta)\)

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{lll} {\text { Conic }} & \text { Eccentricity } & \text { Directrix } \\ \text { Hyperbola } & e=\frac{3}{2} & x=-1 \end{array} $$
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