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Chapter 4: Problem 2

Exer. 1-2: Change to logarithmic form. (a) \(3^{5}=243\) (b) \(3^{-4}=\frac{1}{81}\) (c) \(c^{p}=d\) (d) \(7^{x}=100 p\) (e) \(3^{-2 x}=\frac{P}{F}\) (f) \((0.9)^{t}=\frac{1}{2}\)

Short Answer

Expert verified
Convert each exponential to logarithmic as follows: (a) \( \log_3 243 = 5 \), (b) \( \log_3 \frac{1}{81} = -4 \), (c) \( \log_c d = p \), (d) \( \log_7 (100p) = x \), (e) \( \log_3 \frac{P}{F} = -2x \), (f) \( \log_{0.9} \frac{1}{2} = t \).

Step by step solution

01

Understand the Exercise

The task is to change exponential equations into their equivalent logarithmic form. The general formula for this conversion is: if \( a^b = c \), then the logarithmic form is \( \log_a c = b \). Apply this rule to each sub-part of the exercise.
02

Convert Part (a)

Given the exponential equation \( 3^5 = 243 \), convert it to logarithmic form as \( \log_3 243 = 5 \).
03

Convert Part (b)

Given the exponential equation \( 3^{-4} = \frac{1}{81} \), convert it to logarithmic form as \( \log_3 \frac{1}{81} = -4 \).
04

Convert Part (c)

The given equation is \( c^p = d \). Convert it to logarithmic form as \( \log_c d = p \).
05

Convert Part (d)

The given equation is \( 7^x = 100p \). Convert it into logarithmic form as \( \log_7 (100p) = x \).
06

Convert Part (e)

For the equation \( 3^{-2x} = \frac{P}{F} \), write the logarithmic form as \( \log_3 \frac{P}{F} = -2x \).
07

Convert Part (f)

The given equation is \( (0.9)^t = \frac{1}{2} \). Convert it to logarithmic form as \( \log_{0.9} \frac{1}{2} = t \).

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Key Concepts

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

Exponential Equations
Exponential equations involve expressions where variables are found in the exponent position. In simple terms, they are equations in the format of \( a^b = c \), where \( a \) is the base, \( b \) is the exponent, and \( c \) is the result of raising \( a \) to the power of \( b \). These equations are common in both natural phenomena and mathematical models.
  • They are often used to model growth or decay, such as population growth or radioactive decay.
  • Solving these equations typically involves manipulating the exponents or converting to a different form like logarithmic form for easier handling.
Understanding exponential equations is crucial because of their vast applications in science, technology, engineering, and mathematics (STEM). Grasping the fundamentals allows students to delve deeper into complex topics where these equations arise frequently.
Logarithmic Conversion
Logarithmic conversion is the process of changing an exponential equation into a logarithmic form. This transformation is useful because logarithms can simplify the process of solving exponential equations. The basic transformation rule is: if you have an equation \( a^b = c \), you can convert it to logarithmic form as \( \log_a c = b \).
  • This step is crucial when the exponent (\( b \)) is unknown, as logarithms can be used to isolate and solve for \( b \).
  • Converting to a logarithmic form changes a multiplicative problem into an additive one, often simplifying the overall math.
To better understand it, let's look at concrete examples. If we consider \( 3^5 = 243 \), the logarithmic form \( \log_3 243 = 5 \) indicates that the logarithm base 3 of 243 is 5, providing insight into the power needed on 3 to produce 243.
Precalculus
Precalculus serves as an essential foundation for calculus, combining elements of algebra and trigonometry. It addresses functions and their properties, including exponential and logarithmic functions.
  • Students learn how to manipulate different types of equations, develop graph interpretation skills, and understand the behavior of various functions.
  • A major part of precalculus involves learning to move fluidly between exponentials and logarithms, as these concepts form a core part of calculus studies.
For example, understanding the conversion from an exponential to a logarithmic equation is vital. You leverage skills from precalculus to solve problems in physics, engineering, and even economics, where such conversions can simplify complex models.
Functions and Graphs
Functions describe relationships between sets of data and are often represented graphically. Understanding these graphs helps visualize how changes in one variable affect another. Exponential and logarithmic functions have their own distinctive graphs.
  • Exponential functions form curves that rapidly increase or decrease, depending on whether the base is larger than one or between zero and one.
  • Logarithmic functions, conversely, increase slowly and have a vertical asymptote, often serving as the inverse of exponential functions.
By graphing functions, you can better understand their behavior:
  • Recognize growth and decay patterns immediately.
  • Use graphing as a tool to check your mathematical conversions and solutions.
Mastering these concepts ensures you can transition from algebraic manipulation to practical, real-world applications.

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Most popular questions from this chapter

Exer. \(85-86:\) Approximate the real root of the equation. $$\ln x+x=0$$

In 1840 , Britain experienced a bovine (cattle and oxen) epidemic called epizooty. The estimated number of new cases every 28 days is listed in the table. At the time, the London Daily made a dire prediction that the number of new cases would continue to increase indefinitely. William Farr correctly predicted when the number of new cases would peak. Of the two functions $$\begin{array}{l}f(t)=653(1.028)^{t} \\\g(t)=54,700 e^{-(t-200)^{2} / 7500}\end{array}$$ and one models the newspaper's prediction and the other models Farr's prediction, where \(t\) is in days with \(t=0\) corresponding to August \(12,1840\). $$\begin{array}{|c|c|}\hline \text { Date } & \text { New cases } \\\\\hline \text { Aug. 12 } & 506 \\\\\hline \text { Sept. 9 } & 1289 \\\\\hline \text { Oct. 7 } & 3487 \\\\\hline \text { Nov. 4 } & 9597 \\\\\hline \text { Dec. 2 } & 18,817 \\\\\hline \text { Dec. 30 } & 33,835 \\\\\hline \text { Jan. 27 } & 47,191 \\\\\hline\end{array}$$ (a) Graph each function, together with the data, in the viewing rectangle \([0,400,100]\) by \([0,60,000,10,000]\) (b) Determine which function better models Farr's prediction. (c) Determine the date on which the number of new cases peaked.

Glottochronology is a method of dating a language at a particular stage, based on the theory that over a long period of time linguistic changes take place at a fairly constant rate. Suppose that a language originally had \(N_{0}\) basic words and that at time \(t,\) measured in millennia (1 millennium = 1000 years), the number \(N(t)\) of basic words that remain in common use is given by \(N(t)=N_{0}(0.805)^{t}\). (a) Approximate the percentage of basic words lost every 100 years. (b) If \(N_{0}=200,\) sketch the graph of \(N\) for \(0 \leq t \leq 5\).

Solve the equation graphically. $$x \log x-\log x=5$$

Urban population density An urban density model is a formula that relates the population density \(D\) (in thousands/mi \(^{2}\) ) to the distance \(x\) (in miles) from the center of the city. The formula \(D=a e^{-b x}\) for central density \(a\) and coefficient of decay \(b\) has been found to be appropriate for many large U.S. cities. For the city of Atlanta in 1970 , \(a=5.5\) and \(b=0.10 .\) At approximately what distance was the population density 2000 per square mile?
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