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Chapter 3: Problem 90

Evaluate or simplify each expression without using a calculator. $$\ln e^{7}$$

Short Answer

Expert verified
The simplified expression for \(\ln e^{7}\) is 7

Step by step solution

01

Identify the base of the logarithm and the exponent

The base of the natural logarithm here is \(e\) and the exponent of \(e\) is 7.
02

Apply the property of logarithms

Since the natural logarithm and the exponential function are inverse functions, \(\ln e^{n} = n\), where n is any real number. In this case, \(n = 7\).
03

Simplify the expression

So, \(\ln e^{7} = 7\)

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Key Concepts

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

Properties of Logarithms
Logarithms are fascinating mathematical tools that can simplify complex calculations. One of the core properties involves simplifying expressions using a base. When dealing with logarithms, it is essential to understand these key properties:
  • Product Rule: The logarithm of a product is the sum of the logarithms: \( \log_b (mn) = \log_b m + \log_b n \).
  • Quotient Rule: The logarithm of a quotient is the difference of the logarithms: \( \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n \).
  • Power Rule: The logarithm of a power is the exponent times the logarithm of the base: \( \log_b (m^n) = n \cdot \log_b m \).
In the exercise, the property used is that the natural logarithm and the exponential function are inverse operations. This means that when you have \( \ln e^n \), it simplifies directly to \( n \), as these two functions cancel each other out. This is very effective for simplifying expressions like \( \ln e^7 \), where you can immediately conclude the answer is 7.
Exponential Function
The exponential function is a fundamental part of mathematics, especially when dealing with growth and decay processes. It is defined as \( f(x) = e^x \), where \( e \) (approximately 2.718) is a special mathematical constant, sometimes referred to as Euler's number.
The following are some important characteristics of exponential functions:
  • They exhibit rapid increase or decrease depending on the sign of the exponent.
  • The slope (rate of change) also increases or decreases rapidly.
  • Every exponential function passes through the point \( (0, 1) \) because \( e^0 = 1 \).
In connection with the original exercise, since the question involves \( e^7 \), understanding this function helps us see how the output drastically increases as the exponent grows. The exponential function's inverse relationship with the natural logarithm \( \ln \) is crucial because it simplifies expressions like \( \ln e^7 \) effortlessly.
Inverse Functions
Inverse functions are pairs of functions that "undo" each other. When a function \( f \) and its inverse \( f^{-1} \) are applied successively, they return the original value. For logarithms and exponential functions, this relationship is direct:
  • The exponential function \( e^x \) and the natural logarithm \( \ln x \) are inverses. When you take \( \ln(e^x) \), the result is \( x \), and vice versa, \( e^{\ln x} = x \).
  • This relationship is a powerful computational tool, especially useful in calculus, algebra, and practical real-world applications.
Let's tie this to the original problem: The expression \( \ln e^7 \) directly evaluates to the exponent, which is 7. Understanding inverse functions allows you to simplify much more complex expressions by recalling these inverse relationships.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. My graph of \(f(x)=3 \cdot 2^{x}\) shows that the horizontal asymptote for \(f\) is \(x=3\)

Explain how to solve an exponential equation when both sides can be written as a power of the same base.

From 1970 through \(2010 .\) The data are shown again in the table. Use all five data points to solve Exercises \(70-74\). $$\begin{array}{cc}\hline \begin{array}{c}x, \text { Number of Years } \\\\\text { after } 1969 \end{array} & \begin{array}{c}y, \text { U.S. Population } \\\\\text { (millions) }\end{array} \\ \hline 1(1970) & 203.3 \\\11(1980) & 226.5 \\\21(1990) & 248.7 \\\31(2000) & 281.4 \\\41(2010) & 308.7 \end{array}$$ a. Use your graphing utility's exponential regression option to obtain a model of the form \(y=a b^{x}\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data? b. Rewrite the model in terms of base \(e\). By what percentage is the population of the United States increasing each year?

In Exercises \(125-132,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$5^{x}=3 x+4$$

Solve each equation in Exercises \(146-148 .\) Check each proposed solution by direct substitution or with a graphing utility. $$(\log x)(2 \log x+1)=6$$
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