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Chapter 4: Problem 143

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(x=\frac{1}{k} \ln y,\) then \(y=e^{k x}\)

Short Answer

Expert verified
The statement 'If \(x=\frac{1}{k} \ln y\), then \(y=e^{k x}\)' is True.

Step by step solution

01

Understanding Logarithm and Exponential Relationship

The base of the natural logarithm is the number 'e', and it has the property that \(\ln(e^x) = x\). Therefore, we can rewrite the initial statement \(x= \frac{1}{k}\ln y\) as \(x=\ln(y^{1/k})\). According to the properties of logarithms, we have \(y^{1/k}=e^x\).
02

Substitution and Verification

Next, substitute the right side of the initial statement i.e., \(y=e^{k x}\) into our derived equation. This substitution gives us \(y=e^{k\ln(y^{1/k})}\). From the rule that \(\ln(a^b)=b\ln a\), we get \(y=e^{\ln(y)}\), which simplifies to \(y=y\).
03

Conclusion

The initial statement is true since we end up with the relation \(y=y\) after our simplifications, which is always correct for all y. The statement that if \(x=\frac{1}{k} \ln y\), then \(y=e^{k x}\) is indeed true.

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Most popular questions from this chapter

Research applications of logarithmic functions as mathematical models and plan a seminar based on your group's research. Each group member should research one of the following areas or any other area of interest: \(\mathrm{pH}\) (acidity of solutions), intensity of sound (decibels), brightness of stars, human memory, progress over time in a sport, profit over time. For the area that you select, explain how logarithmic functions are used and provide examples.

Suppose that a population that is growing exponentially increases from \(800,000\) people in 2007 to \(1,000,000\) people in \(2010 .\) Without showing the details, describe how to obtain the exponential growth function that models the data.

We see from the calculator screen at the bottom of the previous page that a logistic growth model for world population, \(f(x),\) in billions, \(x\) years affer 1949 is $$ f(x)=\frac{12.57}{1+4.11 e^{-0.026 x}} $$ Use this function to solve Exercises \(38-42\) How well does the function model the data showing a world population of 6.1 billion for \(2000 ?\)

Writing in Mathematics Nigeria has a growth rate of 0.025 or \(2.5 \% .\) Describe what this means.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can solve \(4^{x}=15\) by writing the equation in logarithmic form.
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