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Chapter 4: Problem 46

$$\text { Find each value. If applicable, give an approximation to four decimal places.}$$ $$\ln e^{5 x}$$

Short Answer

Expert verified
\text{ln}(e^{5x}) = 5x.

Step by step solution

01

Understand the property of logarithms

The natural logarithm function \(\text{ln}(x)\) and the exponential function \(e^x\) are inverse functions. One important property to remember is \(\text{ln}(e^x) = x\). This property will help simplify the given expression.
02

Apply the property to the given expression

Given the expression is \(\text{ln}(e^{5x})\). Using the property \(\text{ln}(e^x) = x\), set \(x = 5x\).
03

Simplify the expression

According to the property \(\text{ln}(e^{5x}) = 5x\). No further calculation is needed as this is the simplified form.

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Key Concepts

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

Inverse Functions
Inverse functions are pairs of functions that undo each other's effect. For instance, if you have a function \(f\) and its inverse \(f^{-1}\), applying \(f\) and then \(f^{-1}\) brings you back to your original value. In simpler terms:
  • \(f(f^{-1}(x)) = x\)
  • \(f^{-1}(f(x)) = x\)
A key example includes the natural logarithm function \(\ln(x)\) and the exponential function \(e^x\). These two functions are inverses because:
  • \(\ln(e^x) = x\)
  • \(e^{\ln(x)} = x\)
Understanding inverse functions is crucial because they help simplify complex expressions, as we saw in the provided exercise.
Logarithms
Logarithms are the inverse operation to exponentiation. The natural logarithm, represented as \(\ln(x)\), is the logarithm to the base \(e\), where e is an irrational constant approximately equal to 2.71828. Logarithms answer the question: 'To what exponent must the base be raised, to yield a given number?'For example, if \(e^y = x\), then \(\ln(x) = y\). Logarithms express the exponents and have several important properties:
  • \(\ln(1) = 0\) because \(e^0 = 1\)
  • \(\ln(e) = 1\) because \(e^1 = e\)
  • \(\ln(x \cdot y) = \ln(x) + \ln(y)\)
  • \(\ln(\frac{x}{y}) = \ln(x) - \ln(y)\)
In the context of the exercise, we used the property \(\ln(e^x) = x\) to simplify \(\ln(e^{5 x})\) to \(5x\).
Exponential Functions
Exponential functions involve variables in the exponent. The most common bases are \(e\) and 10. An exponential function that uses base \(e\) is generally written as \(e^x\).Exponential functions grow quickly; small changes in the variable result in large changes in the value of the function. Here are some key points about exponentials:
  • \(e^0 = 1\)}
  • For any real number \(x\), \(e^x > 0\)
  • \(e^{x+y} = e^x \cdot e^y\)
  • \( (e^x)^y = e^{xy} \)}
In our exercise, recognizing that the natural logarithm \(\ln\) and the exponential function \(e^x\) are inverses allowed us to simplify \(\ln(e^{5x})\) directly to \(5x\), showcasing the practical usage of exponential properties.

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Most popular questions from this chapter

Let \(u=\ln a\) and \(v=\ln b .\) Write each expression in terms of \(u\) and \(v\) without using the In function. $$\ln \frac{a^{3}}{b^{2}}$$

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