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Chapter 12: Problem 83

A student incorrectly reasons that $$\begin{aligned}\log _{b} \frac{1}{x} &=\log _{b} \frac{x}{x x} \\\&=\log _{b} x-\log _{b} x+\log _{b} x=\log _{b} x\end{aligned}$$ What mistake has the student made?

Short Answer

Expert verified
The student's initial step of rewriting \(\frac{1}{x}\) \(\rightarrow \rightarrow -x\is erroneous.

Step by step solution

01

Identify the student's reasoning

The student attempts to simplify \(\text{Logarithm of a fraction}\) by rewriting \(\frac{1}{x}\) as \(\frac{x}{xx}\).\ Their steps are: \(\begin{aligned} \log _{b} \frac{1}{x} &=\log _{b} \frac{x}{xx} \ \&=\log _{b} x -\log _{b} x +\log _{b} x = \log _{b} x \end{aligned}\)
02

Simplify the correct logarithmic expression

Start with the correct logarithmic property for fractions: \(\log_{b}\frac{1}{x} = \log_{b}(1) - \log_{b}(x)\).Since \(\log_{b}(1)\rightarrow 0\). \(\log_{b}\frac{1}{x} = 0 - \log_{b}(x)=\)Thus, \(\log_{b}\frac{1}{x} = -\log_{b}(x)\)
03

Review the student's mistake

The student's mistake is in the incorrect transformation of \(\frac{1}{x} = \frac{x}{xx}\) \is flawed because \frac{x}{xx}\is \ \=\frac{x}{x^{2}}\which further simplifies to \frac{1}{x}\instead of \the intended \frac{x}{xx}\
04

Correct interpretation

Putting it all together: The correct interpretation of \(\frac{1}{x}\is\rightarrow -\log_{b}(x)\)

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Key Concepts

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

logarithms
Logarithms help solve equations where the unknown is an exponent. For a number base \(b\), \text{logarithm base \(b\)} (\text{expression}) gives the power to which \(b\) needs to be raised to get \text{expression}.
For instance, \text{log}_2(8) = 3 simply means \(2^3 = 8\).

Logarithms have essential properties that aid in simplification and solving complex expressions:
  • \text{Product Property}: \text{log}_{b}(xy) = \text{log}_{b}(x) + \text{log}_{b}(y)
  • \text{Quotient Property}: \text{log}_{b}(\frac{x}{y}) = \text{log}_{b}(x) - \text{log}_{b}(y)
  • \text{Power Property}: \text{log}_{b}(x^y)= y \text{log}_{b}(x)
Understanding these properties is critical for accurately simplifying logarithmic expressions and avoiding mistakes.
simplification of fractions
Simplifying fractions is foundational in many areas of math, including logarithms. When simplifying \frac{a}{b}, you aim to express it in its simplest form by dividing both \(a\) and \(b\) by their greatest common divisor (GCD).

In the context of logarithms, the \text{Quotient Property} of logarithms plays a crucial role. For example, the student mistakenly tried to simplify \text{log}_{b}(\frac{1}{x}) incorrectly by rewriting \frac{1}{x} as \frac{x}{xx}. Proper use of the Quotient Property would show:\text{log}_{b}(\frac{1}{x}) = \text{log}_{b}(1) - \text{log}_{b}(x), which simplifies to:0 - \text{log}_{b}(x) = -\text{log}_{b}(x).

Breaking down fractions correctly first makes accurate logarithmic simplification possible.
common logarithmic mistakes
When working with logarithms, it's easy to make mistakes, especially if you neglect the logarithmic properties.
  • Incorrect Fraction Simplification: Just like the student in the exercise mistakenly rewrote \frac{1}{x} as \frac{x}{xx}, remember that \frac{1}{x} is already in its simplest form.
  • Omitting the Base: Always specify the base of a logarithm.When not indicated, the default base is 10, shorthand as \(\text{log}\).If it鈥檚 natural logarithm, it uses base \(e\), notated as \(\text{ln}\).
  • Forgetting Logarithm of 1: \(\text{Log}_{b}(1)\) equals 0 for any base \(b\), because: \(b^0=1\)
To avoid these mistakes, always adhere to the properties, and meticulously simplify fractions.

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Most popular questions from this chapter

As of February \(2010,\) Alex Rodriguez of the New York Yankees had the largest contract in sports history. As part of the 10 -year \(\$ 275\) -million deal, he will receive \(\$ 20\) million in \(2016 .\) How much money would need to be invested in 2008 at \(4 \%\) interest, compounded continuously, in order to have \(\$ 20\) million for Rodriguez in \(2016 ?\) (This is much like determining what \(\$ 20\) million in 2016 is worth in 2008 dollars.

Express as an equivalent expression that is a sum or a difference of logarithms and, if possible, simplify. If \(\log _{a} x=2, \log _{a} y=3,\) and \(\log _{a} z=4,\) what is $$\log _{a} \frac{\sqrt[3]{x^{2} z}}{\sqrt[3]{y^{2} z^{-2}}} ?$$

$$\text { If } \log _{a} x=2, \text { what is } \log _{a}(1 / x) ?$$

Solve for \(x\) $$ \log \left(275 x^{2}\right)=38 $$

Solve. Value of a Sports Card. Legend has it that because he objected to teenagers smoking, and because his first baseball card was issued in cigarette packs, the great shortstop Honus Wagner halted production of his card before many were produced. One of these cards was purchased in 1991 by hockey great Wayne Gretzky (and a partner) for \(\$ 451,000 .\) The same card was sold in 2007 for \(\$ 2.8\) million. For the following questions, assume that the card's value increases exponentially, as it has for many years. a) Find the exponential growth rate \(k,\) and determine an exponential function that can be used to estimate the dollar value, \(V(t),\) of the card \(t\) years after 1991 b) Predict the value of the card in 2012 c) What is the doubling time for the value of the card? d) In what year will the value of the card first exceed \(\$ 4,000,000 ?\)
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