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Chapter 4: Problem 3

Write the logarithmic equation as an exponential equation, or vice versa. $$ \ln 0.2=-1.6094 \ldots $$

Short Answer

Expert verified
The exponential form of the equation \( \ln 0.2=-1.6094 \) is \( e^{-1.6094} = 0.2 \).

Step by step solution

01

Model the Logarithms

Remember, the natural logarithm \(\ln x\) can be understood as \( \ln x = \log_{e} x \) where \( e \) is Euler's number, which roughly equals \(2.71828\). This understanding will be of great help when we convert the equation from logarithmic form to exponential form.
02

Conversion to Exponential Form

The general definition of a logarithm is that if \( b^{y} = x \), then \( \log_{b} x = y \). We can use this pattern to convert our given logarithm \(\ln 0.2=-1.6094\) to its equivalent exponential form. Thus, the corresponding exponential form gets the form \( e^{-1.6094} = 0.2 \).

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Key Concepts

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

Exponential Equation
Exponential equations involve expressions where a constant, called the base, is raised to a variable exponent. These equations have the general form of \( b^y = x \). The base \( b \) is typically a positive number, and the variable \( y \) is what makes the equation exponential.
Exponential equations are solved by isolating the exponent to solve for the variable. This often requires converting between exponential and logarithmic forms.
  • Example: If \( 3^x = 9 \), we recognize that \( 9 \) is \( 3^2 \), so \( x = 2 \).
  • Sometimes, numerical approximations are used, especially when the exponents are not easily simplified.
Natural Logarithm
The natural logarithm is a special type of logarithm with the base \( e \), where \( e \) is approximately 2.71828. It is usually denoted as \( \ln(x) \).
It has unique properties that set it apart from other logarithms, making it a fundamental function in mathematics and real-world applications. The natural logarithm is especially prevalent in fields like calculus, finance, and physics.
  • Property: \( \ln(1) = 0 \) because \( e^0 = 1 \).
  • Property: \( \ln(e) = 1 \) because \( e^1 = e \).
  • Used for solving exponential growth and decay problems.
Using natural logarithms, expressions such as \( e^x \) can be inverted, allowing us to solve for \( x \).
Euler's Number
Euler's number, denoted by \( e \), is an important mathematical constant approximately equal to 2.71828. It is named after the mathematician Leonhard Euler.
Its significance arises from its unique properties in both theoretical and applied mathematics. For example, functions based on \( e \) such as \( e^x \) have derivatives and integrals that are also functions of \( e^x \), making them exponentially relevant in calculus.
  • Used to model growth processes that have continuous growth rates, like populations or investments.
  • The exponential function \( e^x \) and its inverse, the natural logarithm \( \ln(x) \), are the building blocks of complex mathematical concepts.
  • Appears in compound interest calculations, Bernoulli trials, and Euler's formula in complex numbers.
Euler's number is not just a mathematical curiosity, but a practical tool for solving real-world problems.
Conversion of Equations
Converting between logarithmic and exponential equations is a key skill in algebra. Understanding the relationship allows us to solve complex equations more effectively.
To convert a logarithmic equation like \( \log_b(x) = y \) to an exponential equation, you use the exponential form \( b^y = x \). Conversely, converting \( b^y = x \) back to logarithmic form gives \( \log_b(x) = y \).
  • Conversion helps in solving equations where the variable is difficult to isolate in one form.
  • For the natural logarithm of \( \ln(x) = y \), the equivalent exponential form is \( e^y = x \).
  • This conversion is often used in calculus, particularly when dealing with integrals and derivatives of exponential functions.
Mastering this conversion increases your problem-solving toolkit, enabling you to work through exponential growth, decay problems, and beyond.

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