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Chapter 9: Problem 8

Write an equivalent exponential or logarithmic equation. \(\ln 1=0\)

Short Answer

Expert verified
The equivalent exponential equation is \( e^0 = 1 \).

Step by step solution

01

Understanding the Equation

The given equation is \( \ln 1 = 0 \). This is a logarithmic equation which involves the natural logarithm. It expresses the exponent to which the base of the natural logarithm (\( e \)) must be raised to produce the number 1.
02

Writing the Exponential Form

To convert the logarithmic equation into an exponential form, we use the fact that the natural logarithm is the inverse of the exponential function with base \( e \). The general relationship is: if \( \ln a = b \), then \( e^b = a \).
03

Applying the General Rule

Using the rule from Step 2, apply it to the given equation \( \ln 1 = 0 \). This means that \( e^0 = 1 \) because the natural logarithm \( ln \) of 1 is 0, indicating that 1 is \( e \) raised to the power of 0.

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Key Concepts

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

Natural Logarithm
The natural logarithm is a logarithm to the base of the number "e". It's often written as "ln" followed by the number in question. For example, in the equation \( \ln 1 = 0 \), "ln" indicates that we are dealing with a natural logarithm.
The natural logarithm is particularly helpful in continuous compounding situations and is widely used in various fields such as mathematics, physics, and engineering.
  • The natural logarithm of 1 is always 0. This is because any non-zero number to the power of 0 is 1.
  • The notation "ln" implies the base of the logarithm is "e", which is approximately 2.718.
This property makes natural logarithms very useful in solving equations where "e" is involved.
Exponential Form
Converting a logarithmic equation to exponential form is a common technique used to simplify or solve equations. The exponential form of a logarithmic equation converts it into an expression where a base is raised to an exponent.
For example, the equation \( \ln 1 = 0 \) can be rewritten in its exponential form. If you have \( \ln a = b \), then in exponential form, it becomes \( e^b = a \).
  • In the case of \( \ln 1 = 0 \), this becomes \( e^0 = 1 \).
  • This shows the inverse relationship between logarithms and exponents.
Understanding how to move between logarithmic and exponential forms is a key skill in algebra and calculus.
Base e
The base \( e \) is a fundamental mathematical constant, similar to the way \( \pi \) (pi) functions in geometry. This constant is approximately 2.718 and is the base of natural logarithms.
The significance of "e" is it's the unique base rate of growth shared by all continually growing processes. In the context of natural logarithms and exponential functions, "e" serves as the foundation due to its natural properties.
  • Any exponential function with the base "e" is referred to as an exponential function, such as \( e^x \).
  • The number "e" is crucial in many aspects of calculus, particularly in dealing with rates of change and growth.
Because of these properties, "e" is extensively used in both pure and applied mathematics.

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