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Chapter 3: Problem 20

Write each logarithmic equation in its equivalent exponential form. $$z=\ln x^{y}$$

Short Answer

Expert verified
The equivalent exponential form is \( e^z = x^y \).

Step by step solution

01

Understand the Logarithmic Form

The equation is given as \( z = \ln x^y \). The natural logarithm \( \ln a \) represents the exponent to which the base \( e \) must be raised to yield \( a \). Here, \( a = x^y \), so the equation \( z = \ln x^y \) can be rewritten in exponential form.
02

Rewrite in Exponential Form

Recall that \( \ln a = b \) can be written in exponential form as \( e^b = a \). Applying this to the equation \( z = \ln x^y \), we set \( b = z \) and \( a = x^y \). Thus, we write \( e^z = x^y \).

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Key Concepts

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

Understanding Exponential Form
The concept of exponential form is about expressing an equation where a number is raised to a power. In simpler terms, if you have an equation like \( z = \ln x^{y} \),you can convert it from logarithmic form to exponential form. This helps because exponentials are often easier to understand and manipulate in mathematical equations. Recognizing that \( \ln x^{y} = z \) means the natural logarithm of \( x^{y} \) equals \( z \),we can express this as an exponential equation:
  • Identify the base: In natural logarithms, it's always \( e \), which is approximately 2.718.
  • Identify the power: In this equation, it鈥檚 \( z \).
  • Convert it: Rewrite this as \( e^z = x^{y} \).This shows the equivalence between them.
Exponential form reveals the relationship in a more direct way, which can simplify solving and understanding equations.
Exploring Natural Logarithms
Natural logarithms, denoted as \( \ln \), are logarithms with the base \( e \) (about 2.71828). The symbol \( \ln \) comes from the Latin 'logarithmus naturalis'.These logarithms are frequently used in sciences and engineering because they relate to growth processes and things like interest rates. Subtly linking calculations to exponential growth and natural decline, natural logs enable us to work efficiently with phenomena occurring in nature and complex systems.
For example:
  • If \( \ln(x) = 1 \), then \( e^1 = x \), making \( x = e \).
  • When \( \ln \) is used in equations like \( z = \ln x^{y} \), it expresses the power or exponent needed for \( e \) to produce the input number. Here, showing how \( e^z \) represents \( x^y \).
Even complex expressions, like those involving powers, make more sense and are more approachable after translating with logarithms.
Grasping the Idea of Exponents
Exponents dictate how many times a base number is multiplied by itself, which simplifies repeated multiplication. Consider the base \( x \), raised to the exponent \( y \), written as \( x^y \). This means multiply \( x \) by itself \( y \) number of times. Exponents allow compound calculations to be concise and workable.
Key points include:
  • Exponential growth is pervasive in fields like biology and finance, represented naturally in the form.\( x^0 = 1 \) for any \( x \). This rule makes sense because multiplying no factors must logically equal one.
  • Manipulating equations: In \( z = \ln x^y \), recognizing \( x^y \) involves exponents is crucial as it enables us to transform the equation.Exponents contribute immensely to understanding and solving equations!
With these insights, not only can we easily express equations differently, but also decode what these variations mean in practical applications.

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Most popular questions from this chapter

In Exercises 49 and 50 , refer to the logistic model \(f(t)=\frac{a}{1+c e^{-k t}},\) where \(a\) is the carrying capacity. As \(k\) increases, does the model reach the carrying capacity in less time or more time?

Suppose the final exam in this class has a normal, or bell-shaped, grade distribution of exam scores, with an average score of \(80 .\) An approximate function that models your class's grades on the exam is \(N(x)=10 e^{-(x-80)^{2} / 16^{2}},\) where \(N\) represents the number of students who received the score \(x\) a. Graph this function. b. What is the average grade? c. Approximately how many students scored a \(60 ?\) d. Approximately how many students scored \(100 ?\)

Explain the mistake that is made. Solve the equation: \(4 e^{x}=9\) Solution: Take the natural log of both sides. \(\quad \ln \left(4 e^{x}\right)=\ln 9\) Apply the property of inverses. \(4 x=\ln 9\) \(x=\frac{\ln 9}{4} \approx 0.55\) Solve for \(x\) This is incorrect. What mistake was made?

Use a graphing calculator to plot \(y=\ln (2 x)\) and \(y=\ln 2+\ln x .\) Are they the same graph?

In Exercises 49 and 50 , refer to the logistic model \(f(t)=\frac{a}{1+c e^{-k t}},\) where \(a\) is the carrying capacity. As \(c\) increases, does the model reach the carrying capacity in less time or more time?
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