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Chapter 1: Problem 49

Solve for \(y\) in terms of \(t\) or \(x,\) as appropriate. $$\ln y=2 t+4$$

Short Answer

Expert verified
\(y = e^4 \cdot e^{2t}\)

Step by step solution

01

Understand the Equation

We start with the given equation \(\ln y = 2t + 4\). This is a logarithmic equation which suggests that the natural logarithm of \(y\) is equal to the expression \(2t + 4\).
02

Convert to Exponential Form

To eliminate the natural logarithm, we rewrite the equation in its exponential form. Recall that \(\ln y = x\) implies \(y = e^x\). Thus, \(\ln y = 2t + 4\) becomes \(y = e^{2t + 4}\).
03

Simplify the Exponential Expression

The expression \(e^{2t + 4}\) can be further rewritten as \(y = e^4 \, e^{2t}\) because of the property that \(e^{a+b} = e^a \cdot e^b\).
04

Final Expression

Thus, the simplified expression for \(y\) in terms of \(t\) is \(y = e^4 \cdot e^{2t}\). This is the final form of \(y\) expressed in terms of \(t\).

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Key Concepts

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

Exponential Form
When working with logarithmic equations, converting to exponential form is a central technique. This involves changing a logarithmic equation into an exponential one. To put it simply, if you have a natural logarithm like \(\ln y = x\), it can be rewritten in its exponential form as \(y = e^x\).
This conversion leverages the relationship between logs and exponents:
  • Natural logarithms (\(\ln\)) are based on the constant \(e\), which is approximately 2.718.
  • By rewriting the logarithmic equation into exponential form, you isolate the variable \(y\), making it easier to solve or simplify further.
  • Understanding how to transition between these forms is crucial for solving complex equations that involve logarithmic and exponential functions.
Mastering the exponential form helps to unlock new pathways in mathematics where direct solutions might not be easily visible with the initial logarithmic equation.
Natural Logarithm
The natural logarithm, denoted as \(\ln\), is a specific logarithm that uses \(e\) as its base. It is widely used in calculus and mathematical modeling due to its unique properties.
The equation \(\ln y = 2t + 4\) can be interpreted as asking, "To what power must \(e\) be raised to yield \(y\)?"
Some key points about natural logarithms include:
  • \(\ln\) is the inverse function of the exponential function \(e^x\).
  • Natural logarithms are useful in solving real-world problems, like calculating exponential growth or decay.
  • They allow for simplification processes in various equations, as their properties can often reduce complex expressions into more manageable forms.
Recognizing when to apply the natural logarithm, and how to handle it, can effectively transform a challenging problem into an easily solvable one.
Simplification Techniques
The simplification of expressions such as \(e^{2t+4}\) is an essential skill in algebra and pre-calculus. By breaking these down into smaller parts or using well-known mathematical properties, understanding becomes more accessible.
For instance, by applying the property \(e^{a+b} = e^a \cdot e^b\), the expression \(e^{2t+4}\) can be split as follows:
  • \(y = e^4 \cdot e^{2t}\)
  • This illustrates the additive property of exponents, which states that an exponent on a product can be separated into a product of exponents.
  • This simplification helps in further operations, like multiplying or factoring, and provides a more tangible understanding of the relationship between the base \(e\) components.
Simplifying complex expressions helps in navigating mathematical challenges with ease, reducing the risk of mistakes and improving calculation efficiency.

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Most popular questions from this chapter

Evaluate \(\cos \frac{11 \pi}{12}\) as \(\cos \left(\frac{\pi}{4}+\frac{2 \pi}{3}\right)\)

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