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Chapter 3: Problem 17

Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. $$\log _{1 / 2} 4$$

Short Answer

Expert verified
-2.000

Step by step solution

01

Setting Up the Change-of-Base Formula

Apply the change-of-base formula to re-write the equation. Given the equation \( \log _{1 / 2} 4 \), we can use the change-of-base formula so that c is 10 (this is a common base that is often used in logarithms). The change-of-base formula gives: \( \log _{1 / 2} 4 = \frac{\log 4}{\log (1/2)} \).
02

Calculating the Values

Calculate the values of \( \log 4 \) and \( \log (1/2) \). The result is: \( \frac{\log 4}{\log (1/2)} = \frac{0.6020599913}{-0.3010299957} \).
03

Division

Now compute the division. After dividing, the result is -2.
04

Rounding to Three Decimal Places

Round the result to three decimal places. Since the result is -2, and there are no decimal places in this answer, we get -2.000 as the final answer when rounded to three decimal places.

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Key Concepts

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

Understanding Logarithms
Logarithms are a fundamental concept in mathematics used to solve equations involving exponents. A logarithm, simply put, is an operation that helps us determine the power to which a number, called the "base," must be raised to obtain another number. For instance, in the equation \( b^x = n \), \( x \) is the logarithm of \( n \) with base \( b \). We write this as \( \log_{b}(n) = x \).
  • If we say \( \log_{10}(100) = 2 \), we mean 10 raised to the power of 2 is 100.
  • When we deal with different bases, it helps us solve complex problems involving growth, decay, and signal processing among other applications.
The logarithm makes it easier to work with large numbers or powers and is inversely related to exponentiation. When faced with an equation like \( \log_{1/2}(4) \), we can infer this is the exponent needed to make \( 1/2 \) into 4.
Base Conversion Using Change-of-Base Formula
The change-of-base formula is a useful tool for converting a logarithm from one base to another, making calculations easier especially when we use a calculator that typically supports only base 10 (common logarithms) or base \( e \) (natural logarithms). The formula is given by:\[\log_{b}(n) = \frac{\log_{c}(n)}{\log_{c}(b)}\]where \( c \) is the new base.
  • In the original exercise, we have: \( \log _{1 / 2} 4 \) which can be transformed into \( \frac{\log 4}{\log (1/2)} \).
  • By choosing base 10 (\( c = 10 \)) as a common base, we make it compatible with most calculators.
  • This transformation helps simplify complex numbers and operations, translating them into a form more suited for straightforward calculation.
With this approach, identifying accurate values for seemingly complex expressions becomes much more manageable.
Applying Math Problem-Solving Skills
Problem-solving in math isn't just about plugging numbers into a formula; it involves a series of deliberate steps to analyze and solve the problem at hand effectively. Here's how you can apply problem-solving strategies to logarithmic problems:
  • **Understanding the problem:** Determine what you are solving for, such as calculating a logarithm in a different base.
  • **Using the right formula:** Identify the appropriate mathematical tools or formulas, like the change-of-base formula in our case.
  • **Accuracy in calculation:** Be precise with your computation, ensuring you handle negative and positive values correctly. For example, when solving \( \frac{\log 4}{\log (1/2)} \), compute carefully since the logarithm of a fraction could yield negative outcomes.
  • **Cross-check solutions:** Make sure your final result makes sense, such as ensuring the answer is logically rounded to three decimal places as required.
Through practice and application of these skills, one develops a strong foundation in problem-solving, making complex mathematical operations less daunting.

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Most popular questions from this chapter

The graph of a Gaussian model is ________ shaped, where the ________ ________ is the maximum -value of the graph.

A laptop computer that costs $$\$ 1150$$ new has a book value of $$\$ 550$$ after 2 years. (a) Find the linear model \(V=m t+b\). (b) Find the exponential model \(V=a e^{k t}\) (c) Use a graphing utility to graph the two models in the same viewing window. Which model depreciates faster in the first 2 years? (d) Find the book values of the computer after 1 year and after 3 years using each model. (e) Explain the advantages and disadvantages of using each model to a buyer and a seller.

Automobiles are designed with crumple zones that help protect their occupants in crashes. The crumple zones allow the occupants to move short distances when the automobiles come to abrupt stops. The greater the distance moved, the fewer g's the crash victims experience. (One \(g\) is equal to the acceleration due to gravity. For very short periods of time, humans have withstood as much as 40 g's.) In crash tests with vehicles moving at 90 kilometers per hour, analysts measured the numbers of g's experienced during deceleration by crash dummies that were permitted to move \(x\) meters during impact. The data are shown in the table. A model for the data is given by \(y=-3.00+11.88 \ln x+(36.94 / x),\) where \(y\) is the number of g's. $$ \begin{array}{|c|c|} \hline x & \text { g's } \\ \hline 0.2 & 158 \\ 0.4 & 80 \\ 0.6 & 53 \\ 0.8 & 40 \\ 1.0 & 32 \\ \hline \end{array} $$ (a) Complete the table using the model. $$ \begin{array}{|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\ \hline y & & & & & \\ \hline \end{array} $$ (b) Use a graphing utility to graph the data points and the model in the same viewing window. How do they compare? (c) Use the model to estimate the distance traveled during impact if the passenger deceleration must not exceed \(30 \mathrm{~g}\) 's. (d) Do you think it is practical to lower the number of g's experienced during impact to fewer than \(23 ?\) Explain your reasoning.

Use the acidity model given by \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right],\) where acidity \((\mathrm{pH})\) is a measure of the hydrogen ion concentration \(\left[\mathrm{H}^{+}\right]\) (measured in moles of hydrogen per liter) of a solution. Find the \(\mathrm{pH}\) if \(\left[\mathrm{H}^{+}\right]=2.3 \times 10^{-5}\).

$$\$ 2500$$ is invested in an account at interest rate \(r\), compounded continuously. Find the time required for the amount to (a) double and (b) triple. $$r=0.0375$$
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