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Chapter 3: Problem 7

Exercises \(7-10\), rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. $$\log _{5} 16$$

Short Answer

Expert verified
The logarithm \(\log_{5} 16\) rewritten as a ratio of common logarithms is \(\frac{\log_{10} 16}{\log_{10} 5}\) and as a ratio of natural logarithms is \(\frac{\ln 16}{\ln 5}\)

Step by step solution

01

Write down the problem

The exercise provides the following logarithm: \(\log_5 16\) that needs to be rewritten as a ratio of common and natural logarithms
02

Convert to common logarithms

We will first convert the given logarithm into a ratio of common logarithms (base 10). We'll use the formula \(\log_b a = \frac{\log_c a}{\log_c b}\) where \(c\) is 10 (indicating a common logarithm). Applying this, we get \(\log_{5} 16 = \frac{\log_{10} 16}{\log_{10} 5}\)
03

Convert to natural logarithms

Next, we'll convert the given logarithm into a ratio of natural logarithms (base e). Again, we'll use the same formula, with \(c\) being \(e\). This yields \(\log_{5} 16 = \frac{\log_{e} 16}{\log_{e} 5}\) or with the common logarithmic notation this equation can be rewritten as \(\log_{5} 16 = \frac{\ln 16}{\ln 5}\)

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Key Concepts

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

Common Logarithms
Common logarithms are logarithms that have a base of 10. They are called "common" because they are frequently used in everyday calculations and scientific reports. In mathematics and science, common logarithms simplify the process of dealing with large numbers and are often used since they align with the decimal system.
  • A common logarithm is written as \ \( \log_{10} x \ \), but we usually drop the base and just write \ \( \log x \ \).
  • They are particularly useful in solving equations where the exponentiation can be cumbersome without a calculator.
Converting any other logarithm into common logarithms can be done using the change of base formula. For example, if you have \ \( \log_{5} 16 \ \), you can express it using common logarithms as \ \( \frac{\log 16}{\log 5} \ \). This allows you to calculate or simplify expressions using a calculator that primarily handles base 10 calculations.
Natural Logarithms
Natural logarithms use the base \( e \), a mathematical constant approximately equal to 2.71828. They are denoted by \( \ln x \) rather than \( \log_e x \), though both notations mean the same thing. Natural logarithms have advantages in calculus and mathematical modeling, especially in growth processes and compounding events.
  • Every natural logarithm can be represented as \ \( \ln x = \log_e x \ \).
  • They commonly appear in formulas related to exponential growth and decay, like compound interest or population models.
To convert other logarithms to natural logarithms, the same change of base formula applies. For instance, transforming \ \( \log_{5} 16 \ \) into natural logarithms yields \ \( \frac{\ln 16}{\ln 5} \ \). This conversion is beneficial when dealing with exponential equations or functions that naturally align with the base \( e \).
Change of Base Formula
The change of base formula is essential when you need to calculate logarithms with a base that is not readily available, like base 5 or base 7, and you wish to switch to a more convenient base, such as 10 or \( e \).
  • The formula is \ \( \log_b a = \frac{\log_c a}{\log_c b} \ \), where \ \( c \ \) can be any base.
  • It essentially states that any logarithm can be rewritten as a fraction of two logarithms of another base, simplifying calculations.
Using this formula, you can effortlessly change any logarithmic base. For example, converting \ \( \log_{5} 16 \ \) to either common or natural logarithms is simple. As shown in the solution, you can represent it as \ \( \frac{\log_{10} 16}{\log_{10} 5} \ \) or \ \( \frac{\ln 16}{\ln 5} \ \). Understanding this conversion helps in making more complex mathematical processes accessible.

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Most popular questions from this chapter

In Exercises \(15-20,\) use the properties of logarithms to rewrite and simplify the logarithmic expression. $$\ln \frac{6}{e^{2}}$$

Population Growth A conservation organization released 100 animals of an endangered species into a game preserve. The preserve has a carrying capacity of 1000 animals. The growth of the pack is modeled by the logistic curve $$p(t)=\frac{1000}{1+9 e^{-0.1656 t}}$$ where \(t\) is measured in months (see figure). (Figure can't copy) (a) Estimate the population after 5 months. (b) After how many months is the population \(500 ?\) (c) Use a graphing utility to graph the function. Use the graph to determine the horizontal asymptotes, and interpret the meaning of the asymptotes in the context of the problem.

Fill in the blanks. Polynomial and rational functions are examples of ___ functions.

Graphical Analysis Use a graphing utility to graph \(f\) and \(g\) in the same viewing window and determine which is increasing at the greater rate as \(x\) approaches + \(\infty\). What can you conclude about the rate of growth of the natural logarithmic function? (a) \(f(x)=\ln x, \quad g(x)=\sqrt{x}\) (b) \(f(x)=\ln x, \quad g(x)=\sqrt[4]{x}\)

A cup of water at an initial temperature of \(78^{\circ} \mathrm{C}\) is placed in a room at a constant temperature of \(21^{\circ} \mathrm{C} .\) The temperature of the water is measured every 5 minutes during a half-hour period. The results are recorded as ordered pairs of the form \((t, T),\) where \(t\) is the time (in minutes) and \(T\) is the temperature (in degrees Celsius). \(\left(0,78.0^{\circ}\right),\left(5,66.0^{\circ}\right),\left(10,57.5^{\circ}\right),\left(15,51.2^{\circ}\right)\) \(\left(20,46.3^{\circ}\right),\left(25,42.4^{\circ}\right),\left(30,39.6^{\circ}\right)\) (a) The graph of the model for the data should be asymptotic with the graph of the temperature of the room. Subtract the room temperature from each of the temperatures in the ordered pairs. Use a graphing utility to plot the data points \((t, T)\) and \((t, T-21)\) (b) An exponential model for the data \((t, T-21)\) is given by \(T-21=54.4(0.964)^{t} .\) Solve for \(T\) and graph the model. Compare the result with the plot of the original data. (c) Take the natural logarithms of the revised temperatures. Use the graphing utility to plot the points \((t, \ln (T-21))\) and observe that the points appear to be linear. Use the regression feature of the graphing utility to fit a line to these data. This resulting line has the form \(\ln (T-21)=a t+b\) Solve for \(T,\) and verify that the result is equivalent to the model in part (b). (d) Fit a rational model to the data. Take the reciprocals of the \(y\) -coordinates of the revised data points to generate the points $$\left(t, \frac{1}{T-21}\right)$$ Use the graphing utility to graph these points and observe that they appear to be linear. Use the regression feature of the graphing utility to fit a line to these data. The resulting line has the form $$\frac{1}{T-21}=a t+b$$ (e) Why did taking the logarithms of the temperatures lead to a linear scatter plot? Why did taking the reciprocals of the temperatures lead to a linear scatter plot?
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