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Chapter 8: Problem 30

In \(21-32,\) for each given logarithm, find \(x,\) the antilogarithm. Write the answer to four decimal places. $$ \ln x=-1.1544 $$

Short Answer

Expert verified
The value of \( x \) is approximately 0.3155.

Step by step solution

01

Understand the the relationship

The given equation involves a natural logarithm, denoted as \( \ln \). Recall that when \( \ln(x) = a \), the expression implies that \( x = e^a \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828. Our task is to express \( x \) in terms of \( e \) raised to the power of -1.1544.
02

Apply the formula

Since \( \ln(x) = -1.1544 \), it follows that \( x = e^{-1.1544} \). We need to calculate this expression to find the value of \( x \).
03

Compute the exponential expression

Using a calculator, compute \( e^{-1.1544} \). Ensure your calculator is set to use natural logarithms, and enter \( e^{-1.1544} \) to find \( x \).
04

Round to four decimal places

The computed value of \( e^{-1.1544} \) is approximately 0.3155. Ensure to round this value to four decimal places to meet the problem requirements.

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Key Concepts

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

Natural Logarithm
In mathematics, the natural logarithm, often expressed as \( \ln \), is a logarithm to the base \( e \). The base \( e \) is a constant approximately equal to 2.71828. The natural logarithm has several important properties and applications in mathematical analysis and real-life problems.
  • It is used to calculate the time required for an investment to grow to a certain amount at a given rate in economics.
  • In calculus, the derivative of \( \ln(x) \) is \( \frac{1}{x} \).
  • The natural logarithm helps make multiplicative processes additive, simplifying calculations.
When you see \( \ln(x) = a \), it means if you raise \( e \) to the power of \( a \), you'll get \( x \). This is crucial when solving equations involving logarithms, like in the given exercise where \( \ln(x) = -1.1544 \), hence \( x = e^{-1.1544} \).
Exponential Function
The exponential function is a powerful mathematical expression, denoted by \( e^x \), where \( e \) is the mathematical constant approximately equal to 2.71828. It's a crucial function in calculus because it uniquely defines its own derivative, meaning the rate of change of the exponential function is proportional to its value.
  • It is used in modeling growth processes, such as populations or radioactive decay.
  • It's also applied in calculating compound interest in finance.
  • In differential equations, it helps solve forms that describe many natural phenomena.
For solving problems like the original exercise, remember the equation \( x = e^a \) when dealing with natural logarithms. It transforms the logarithmic equation \( \ln(x) = -1.1544 \) to an exponential form, where \( x = e^{-1.1544} \). This transformation is key in converting logarithmic problems into a form easily solvable with calculators.
e Constant
The number \( e \) is a mathematical constant that is the base of the natural logarithm. It is approximately equal to 2.71828 and plays a significant role in various fields of science and mathematics.
  • It serves as the base of natural logarithms, helping streamline exponential growth calculations.
  • In finance, \( e \) is fundamental in calculating continuous compounding interest.
  • In calculus, \( e^x \) is important because its derivative is \( e^x \), demonstrating exponential growth or decay.
In the context of the given exercise, \( e \) helps transition from the natural logarithm expression \( \ln(x) = -1.1544 \) to its exponential form, where \( x = e^{-1.1544} \). Understanding \( e \) and its application can demystify the concept of logarithmic and exponential functions.

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Most popular questions from this chapter

Solve for \(x : \log _{100} 10=\log _{16} x\)

In \(15-20,\) evaluate each logarithm to the nearest hundredth. $$ \ln \frac{1}{2} $$

In \(48-55,\) if \(\log a=c,\) express each of the following in terms of \(c\) $$ \log \frac{100}{a} $$

In \(48-55,\) if \(\log a=c,\) express each of the following in terms of \(c\) $$ \log a^{2} $$

In \(21-32,\) for each given logarithm, find \(x,\) the antilogarithm. Write the answer to four decimal places. $$ \ln x=-0.5373 $$
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