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Chapter 6: Problem 4

Write an equivalent equation in logarithmic form. a. \(N=10^{-t / c}\) c. \(e^{3 x}=27\) b. \(I=I_{0} \cdot e^{-k / x}\) d. \(\frac{1}{2}=e^{-k t}\)

Short Answer

Expert verified
a. \log_{10} (N) = -t/c; b. \ln(27)=3x; c. \ln(I/I_0)=-k/x; d. \ln(1/2)=-kt

Step by step solution

01

Step 1a - Identify exponential equation

The given equation is: \[ N = 10^{-t / c} \]
02

Step 2a - Apply logarithmic form

Rewrite the equation in logarithmic form. The general form for changing a base-10 exponential equation is \[ a = 10^b \] which converts to \[ \log_{10} (a) = b \]. Apply this to our equation: \[ N = 10^{-t/c} \], becomes \[ \log_{10} (N) = -\frac{t}{c} \]
03

Step 1b - Identify exponential equation

The given equation is: \[ e^{3x} = 27 \]
04

Step 2b - Apply logarithmic form

Rewrite the equation in logarithmic form. The general form for changing a base-e exponential equation is \[ a = e^b \] which converts to \[ \ln (a) = b \]. Apply this to our equation: \[ e^{3x} = 27 \], becomes \[ \ln (27) = 3x \]
05

Step 1c - Identify exponential equation

The given equation is: \[ I = I_0 \cdot e^{-k/x} \]
06

Step 2c - Apply logarithmic form

Rewrite the equation in logarithmic form. The general form for changing a base-e exponential equation is \[ a = e^b \] which converts to \[ \ln (a) = b \]. Apply this to our equation: \[ I = I_0 e^{-k/x} \], becomes \[ \ln (I/I_0) = -\frac{k}{x} \]
07

Step 1d - Identify exponential equation

The given equation is: \[ \frac{1}{2} = e^{-kt} \]
08

Step 2d - Apply logarithmic form

Rewrite the equation in logarithmic form. The general form for changing a base-e exponential equation is \[ a = e^b \] which converts to \[ \ln (a) = b \]. Apply this to our equation: \[ \frac{1}{2} = e^{-kt} \], becomes \[ \ln (\frac{1}{2}) = -kt \]

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Key Concepts

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

Exponential Equations
An exponential equation is one in which the variable appears in the exponent. For instance, the equation \(N=10^{-t / c}\), involves the exponent \(-t/c\). Exponential equations are commonly seen in natural sciences and finance due to their nature of modeling growth or decay.
An important aspect of these equations is understanding how they can be algebraically handled, which includes converting them to logarithmic form for easier manipulation. Knowing how to go from exponential to logarithmic form can simplify solving such equations.
Remember that exponential equations come in different bases - some are based on 10, others on \(e\), the natural logarithm base.
Logarithms
Logarithms are the inverse operations of exponentiation. They answer the question 'to what exponent must we raise a specific base to obtain a given number?' For example, in the base-10 logarithm \(\log_{10}(N) = -\frac{t}{c}\), you are finding the power to which 10 must be raised to get \(N\).
Key points to remember about logarithms include:
  • They convert multiplication into addition.
  • They have properties like \log(ab) = \log(a) + \log(b)\ and \log(a^b) = b \log(a)\.
These properties are extremely useful in simplifying complex equations.
Understanding logarithms is crucial in many fields, including computing and engineering.
Natural Logarithms
Natural logarithms use \(e\) as their base. The constant \(e\) is approximately equal to 2.71828 and is a key number in mathematics, especially in calculus. Natural logs are often denoted as \ln\.
For example, in the equation \(e^{3 x}=27\), converting it to logarithmic form gives \ln(27) = 3x\. This transformation makes it straightforward to solve for \(x\).
Some specific points to remember about \ln\:
  • \ln(e^b) = b\.
  • \ln(1) = 0\.
  • \ln(e) = 1\.
Natural logarithms play an essential role in solving differential equations and modeling continuous growth processes, making them a vital tool in various scientific disciplines.
Base-10 Logarithms
Base-10 logarithms, often denoted as \log\ or \log_{10}\, are used primarily in fields such as engineering and computer science. They simplify the process of working with large numbers by turning multiplication into addition.
For instance, take the equation \(N=10^{-t / c}\). By converting to logarithmic form, we get \log(N) = -\frac{t}{c}\, making it easier to isolate and solve for \(t\) or \(c\).
Key properties of base-10 logarithms include:
  • \log_{10}(10^b) = b\.
  • \log_{10}(100) = 2\ since \100 = 10^2\.
  • \log_{10}(1) = 0\.
These logarithms are especially useful in areas where scaling and orders of magnitude are important, such as seismic activity measurement and sound intensity levels.

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Most popular questions from this chapter

(Requires technology to find a best fit exponential.) According to another version of Moore's Law, the computing power built into chips doubles every 18 months (see Section 5.7 ). The accompanying table shows the computing power of some Intel chips (measured in calculations per second) between 1993 and 2005 . a. Graph the data points (if possible, on a semi-log plot). Explain why an exponential function would be an appropriate model. \begin{tabular}{llr} \hline & & Chip Computing Power (millions of calculations \\ Year & Chip Type & per second) \\ \hline 1993 & Pentium (B) & 66 \\ 1997 & Pentium II & 525 \\ 1999 & Pentium III & 1,700 \\ 2000 & Pentium 4 & 3,400 \\ 2005 & Dual-core Itanium 2 & 27,079 \\ \hline \end{tabular} Source: Intel Corporation. b. Construct an exponential function to model the data in the following two different ways. In each case let \(t=\) number of years since 1993 . i. Using the doubling time given by Moore's Law, construct an exponential function, \(P_{1}(t),\) for chip computing power. ii. Use technology to find a best-fit exponential function, \(P_{2}(t),\) to the data in the table. Does this model come close to verifying Moore's Law \(\left(P_{1}(t)\right) ?\)

The half-life of bismuth-214 is about 20 minutes. a. Construct a function to model the decay of bismuth- 214 over time. Be sure to specify your variables and their units. b. For any given sample of bismuth- 214 , how much is left after I hour? c. How long will it take to reduce the sample to \(25 \%\) of its original size? d. How long will it take to reduce the sample to \(10 \%\) of its original size?

A city of population 1.5 million is expected to experience a \(15 \%\) decrease in population every 10 years. a. What is the 10 -year decay factor? What is the yearly decay factor? The yearly decay rate? b. Use part (a) to create an exponential population model \(g(t)\) that gives the population (in millions) after \(t\) years. c. Create an exponential population model \(h(t)\) that gives the population (in millions) after \(t\) years, assuming a \(1.625 \%\) continuous yearly decrease. d. Compare the populations predicted by the two functions after 20 years. What can you conclude?

Contract, using the rules of logarithms, and express your answer as a single logarithm. a. \(3 \log K-2 \log (K+3)\) b. \(-\log m+5 \log (3+n)\) c. \(4 \log T+\frac{1}{2} \log T\) d. \(\frac{1}{3}(\log x+2 \log y)-3(\log x+2 \log y)\)

For each of the following, find the half-life, then rewrite each function in the form \(P=P_{0} e^{r t}\). Assume \(t\) is measured in years. a. \(P=P_{0}\left(\frac{1}{2}\right)^{t / 10}\) b. \(P=P_{0}\left(\frac{1}{2}\right)^{t / 215}\) c. \(P=P_{0}\left(\frac{1}{2}\right)^{4 t}\)
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