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Chapter 4: Problem 5

If \(\ln x=-1\), write \(x\) using the exponential function.

Short Answer

Expert verified
The value of \( x \) is \( \frac{1}{e} \).

Step by step solution

01

Identify the Given Information

The given equation is \(\ln x = -1\). This equation involves the natural logarithm, which is the logarithm to the base e.
02

Rewrite Using Exponential Form

To solve for \(x\), rewrite the logarithmic equation in its exponential form. The property \(\ln a = b \) can be rewritten as \(a = e^b\).
03

Apply Exponential Form

Using the property, rewrite \( x = -1 \) as \( x = e^{-1} \).
04

Simplify the Expression

Simplify the expression \( e^{-1}\) to \( \frac{1}{e} \). Therefore, \( x = \frac{1}{e} \).

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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 special type of logarithm that has the base of Euler's number, denoted as 'e', which is approximately equal to 2.71828. The natural logarithm is written as \(\backslash \text{ln}\), and it has a unique property that makes it quite useful in mathematics: it can simplify the multiplication and division of exponentials.
Exponential Form
One of the key properties of logarithms is that they can be converted into exponential form. In the context of natural logarithms, if \(\backslash ln a = b\), it means that \(\backslash e^b = a\). This relationship allows you to solve logarithmic equations by rewriting them in an exponential format.
For example, if you have \(\backslash ln x = -1\), you can rewrite it as \(\backslash x = e^{-1}\).
Properties of Logarithms
Logarithms possess several properties that make them powerful tools in mathematics. Some important properties of logarithms include:
  • Product Rule: \(\backslash ln(ab) = ln(a) + ln(b)\).
  • Quotient Rule: \(\backslash ln \frac{a}{b} = ln(a) - ln(b)\).
  • Power Rule: \(\backslash ln(a^b) = b \backslash ln(a)\).
These properties allow you to manipulate and simplify complex logarithmic expressions effectively.
Simplifying Expressions
When working with exponential functions and logarithms, it's crucial to simplify expressions to their simplest form. In the example where we have \(\backslash ln x = -1\) and convert it to exponential form, we get \(\backslash x = e^{-1}\).
The simplified expression of \(\backslash e^{-1}\) is \(\backslash \frac{1}{e}\).
Simplifying expressions helps in understanding and solving mathematical problems more effectively.

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Most popular questions from this chapter

Use logarithmic differentiation to differentiate the following functions. \(f(x)=(x+1)^{4}(4 x-1)^{2}\)

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Solve the given equation for \(x .\) \(\ln x^{4}-2 \ln x=1\)

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