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Chapter 5: Problem 17

Find each of the following. Do not use a calculator. $$\cdot \ln 1$$

Short Answer

Expert verified
\ln(1) = 0

Step by step solution

01

Understand the natural logarithm

The natural logarithm, represented by \( \ln(x) \), is the logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828.
02

Identify the property of the natural logarithm

One of the key properties of the natural logarithm is that \( \ln(1) = 0 \). This is because \( e^0 = 1 \).
03

Use the property

By applying the property mentioned, we find that: \( \ln(1) = 0 \).

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Key Concepts

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

logarithmic properties
To truly understand logarithms, it's essential to grasp their unique properties. Logarithms, including the natural logarithm denoted by \( \ln(x) \), simplify multiplication and exponentiation into addition and subtraction.
This is due to their logarithmic properties, such as:
  • \( \ln(ab) = \ln(a) + \ln(b) \)
  • \( \ln(\frac{a}{b}) = \ln(a) - \ln(b) \)
  • \( \ln(a^n) = n \cdot \ln(a) \)
These rules make it easier to handle complex calculations by transforming them into basic arithmetic operations.
In the given exercise, we use the property of natural logarithms: \( \ln(1) = 0 \). This arises because any number raised to the power 0 is 1, including the constant \( e \). Hence, \( \ln(1) \) is 0, aligning perfectly with property #3.
e constant
The constant \( e \) plays a significant role in mathematics, particularly in natural logarithms and exponential functions. It is an irrational number, approximately equal to 2.71828.
\( e \) was discovered in the context of compound interest calculations and has since found applications in various mathematical fields.
Some key properties of \( e \) include:
  • \( e^1 = e \)
  • \( e^0 = 1 \), which is why \( \ln(1) = 0 \)
  • It is the base of natural logarithms: \( \ln(e) = 1 \).
By understanding \( e \) and its properties, you can easily navigate problems involving natural logarithms and exponential growth or decay.
exponential functions
Exponential functions are crucial in many scientific and mathematical applications. They are expressed as \( f(x) = a e^{bx} \), where \( a \) and \( b \) are constants.
These functions exhibit rapid growth or decay, depending on whether \( b \) is positive or negative.
Some properties of exponential functions include:
  • They always pass through the point (0,1) if not modified by parameters \( a \) or \( b \)
  • Their derivatives are proportional to the function itself: \( f'(x) = b a e^{bx} \)
  • They model a variety of real-world processes like population growth, radioactive decay, and interest calculations
Understanding these properties helps in solving complex problems and applying exponential functions to real-world scenarios.
For example, in the given exercise, recognizing that \( e^0 = 1 \) aids in understanding why \( \ln(1) = 0 \).

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Most popular questions from this chapter

Determine whether each of the following is true or false. Assume that \(a, x, M,\) and \(N\) are positive. $$\frac{\log _{a} M}{x}=\log _{a} M^{1 / x}$$

Find the \(x\) -intercepts and the zeros of the function. $$h(x)=x^{3}-3 x^{2}+3 x-1[4.3]$$

Use a graphing calculator to find the approximate solutions of the equation. $$\ln x^{2}=-x^{2}$$

Consider quadratic functions ( \(a\) )-( h ) that follow. Without graphing them, answer the questions below. a) \(f(x)=2 x^{2}\) b) \(f(x)=-x^{2}\) c) \(f(x)=\frac{1}{4} x^{2}\) d) \(f(x)=-5 x^{2}+3\) e) \(f(x)=\frac{2}{3}(x-1)^{2}-3\) f) \(f(x)=-2(x+3)^{2}+1\) g) \(f(x)=(x-3)^{2}+1\) h) \(f(x)=-4(x+1)^{2}-3\) Which functions have a maximum value?

Determine whether each of the following is true or false. Assume that \(a, x, M,\) and \(N\) are positive. $$\log _{a} M-\log _{a} N=\log _{a} \frac{M}{N}$$
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