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Chapter 11: Problem 31

Write the expressions in Exercises \(31-33\) in the form \(\log _{b} x\) and state the values of \(b\) and \(x\). Verify your answers using a calculator as in Example 6 . $$ \frac{\log 12}{\log 2} $$

Short Answer

Expert verified
Question: Rewrite the expression \(\frac{\log 12}{\log 2}\) in the form \(\log _{b} x\) and find the values of \(b\) and \(x\). Answer: The expression can be rewritten as \(\log_{2} 12\), with \(b=2\) and \(x=12\).

Step by step solution

01

Rewrite the expression using Change of Base Formula

To rewrite the expression in the form \(\log _{b} x\), we need to use the change of base formula, which states that \(\log_{b}x = \frac{\log x}{\log b}\). In our case, we have \(\frac{\log 12}{\log 2}\). Using the change of base formula, we can rewrite this expression as \(\log_{2} 12\). So, the values of \(b\) and \(x\) are 2 and 12, respectively.
02

Verify the answer using a calculator

To verify the answer, we'll calculate \(\log_{2} 12\) using a calculator and compare it with the result of \(\frac{\log 12}{\log 2}\) calculated using a calculator as well. Using a calculator, we find that: - \(\log_{2} 12 \approx 3.584962500721156\) - \(\frac{\log 12}{\log 2} \approx 3.584962500721156\) Since the values are equal, our answer is correct. The expression can be rewritten as \(\log_{2} 12\) with \(b=2\) and \(x=12\).

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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 an important tool in the study of logarithms. It allows us to rewrite logarithms in a different base as a fraction involving common logarithms or natural logarithms. The formula is expressed as:\[\log_{b}x = \frac{\log x}{\log b}\]This formula is useful when you want to evaluate a logarithm with a base that is not readily available on calculators, which typically only have base 10 (common log) or base \(e\) (natural log). In our example, the expression \(\frac{\log 12}{\log 2}\) uses the change of base formula to convert it into \(\log_{2} 12\). Here:
  • \(b = 2\)
  • \(x = 12\)
Using this method helps simplify calculations and enables verification using calculators.
Logarithmic Functions
Logarithmic functions are mathematical expressions that represent the inverse of exponential functions. These functions are written in the form \(\log_{b}(x)\), where \(b\) is the base and \(x\) is the number you want to find the logarithm of. They answer the question: "To what power must the base \(b\) be raised to produce \(x\)?"Logarithmic functions have some key properties:
  • The domain includes all positive real numbers \(x > 0\).
  • The range is all real numbers.
  • The base \(b\) must be a positive number different from 1.
In our example, \(\log_{2} 12\) asks, "What power must 2 be raised to in order to get 12?" Understanding how these functions work is crucial for solving problems involving exponential growth or decay. The study of these functions also aids in grasping logarithmic scales used in various scientific fields.
Calculating Logarithms
Calculating logarithms involves determining the power to which a base number must be raised to obtain another number. This can often be done using a calculator with built-in log functions for bases 10 and \(e\). For other bases, we typically use the change of base formula.Using a calculator:- For \(\log_{2} 12\), since many calculators do not directly provide base 2 logarithms, employ the change of base formula: \(\log_{2}12 = \frac{\log 12}{\log 2}\).- Both terms \(\log 12\) and \(\log 2\) are calculated using common logarithms (base 10), resulting in the same approximate value of \(3.584962500721156\).Always ensure the calculator settings are correct, and that you follow the order of operations when inputting expressions. Using calculators allows precise calculations, crucial for verifying your manual solutions. Whether tackling homework or preparing for exams, mastering the calculation of logarithms strengthens your mathematical foundation.

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Most popular questions from this chapter

Solve the equations, first approximately, as in Example \(1,\) by filling in the given table, and th猫n to four decimal places by using logarithms. $$ \begin{aligned} &\begin{array}{l} \text { Table 11.6 } \text { Solve } 2^{x}=20 \\ \hline \end{array}\\\ &\begin{array}{c|c|c|c|c} \hline x & 4.1 & 4.2 & 4.3 & 4.4 \\ \hline 2^{x} & & & & \\ \hline \end{array} \end{aligned} $$

Problems \(40-43\) concern the Krumbein phi \((\phi)\) scale of particle size, which geologists use to classify soil and rocks, defined by the formula \({ }^{15}\) $$ \phi=-\log _{2} D $$ where \(D\) is the diameter of the particle in \(\mathrm{mm}\). Some particles of clay have diameter \(0.0035 \mathrm{~mm}\). What do they measure on the \(\phi\) scale?

Iodine-131, used in medicine, has a half-life of 8 days. (a) If \(5 \mathrm{mg}\) are stored for a week, how much is left? (b) How many days does it take before only \(1 \mathrm{mg}\) remains?

In \(1936,\) researchers at the Wright-Patterson Air Force Base found that each new aircraft took less time to produce than the one before, owing to what is now known as the learning curve effect. Specifically, researchers found that \(^{16}\) $$ \begin{aligned} f(n) &=t_{0} n^{\log _{2} k} \\ \text { where } f(n) &=\text { Time required to produce } n^{\text {th }} \text { unit } \end{aligned} $$ \(t_{0}=\) Time required to produce the first unit. The value of \(k\) will vary from industry to industry, but for a particular industry it is often a constant. (a) Suppose for a particular industry, the first unit takes 10,000 hours of labor to produce, so \(n_{0}=\) \(10,000,\) and that \(k=0.8\). The learning-curve effect states that the additional hours of labor goes down by a fixed percentage each time production doubles. Show that this is the case by completing the table. By what percent does the required time drop when production is doubled? $$ \begin{array}{c|c|c|c|c|c} \hline n & 1 & 2 & 4 & 8 & 16 \\ \hline f(n) & & & & & \\ \hline \end{array} $$ (b) What kind of function (exponential, power, logarithmic, other) is \(f(n) ?\) Discuss. (c) Suppose a different industry has a lower value of \(t_{0}\), say, \(t_{0}=7000\), but a higher value of \(b\), say, \(b=0.9 .\) Explain what this tells you about the difference in the learning-curve effect between these two industries. (d) What would it mean for \(k>1\) in terms of production time? Explain why this is unlikely to be the case.

Are the expressions in Problems equivalent for positive \(a\) and \(b\) ? If so, explain why. If not, give values for \(a\) and \(b\) that lead to different values for the two expressions. $$ 10^{\log (a+b)} \text { and } a+b $$
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