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Chapter 4: Problem 3

Estimate the indicated value without using a calculator. $$ \ln 0.993 $$

Short Answer

Expert verified
The estimation of \(\ln 0.993\) can be found by using the approximation \(\ln(1+x) \approx x\). Rewrite 0.993 as \(1 + (-0.007)\). Then, \(\ln 0.993 \approx -0.007\).

Step by step solution

01

Determine x for ln(1+x)

We need to write the given number, 0.993, in the form of \(1+x\), so let's find the value of \(x\): \(0.993 = 1 + x \) Solving for x: \(x = 0.993 - 1\) \(x = -0.007\) Now we have written 0.993 as \(1 - 0.007\).
02

Use the approximation ln(1+x) 鈮� x

Now that we have our \(x\) value as -0.007, we can use the approximation \(\ln(1+x) \approx x\) to estimate the natural logarithm of 0.993: \(\ln(1 + (-0.007)) \approx -0.007\)
03

Present the approximation of the natural logarithm

Using the approximation, we find that \(\ln 0.993\) is approximately equal to -0.007.

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Key Concepts

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

Understanding Natural Logarithm
The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \approx 2.71828 \). It is widely used in mathematics to solve equations involving exponential growth, decay, or other processes that change at a rate proportional to their current value.
For example, when you see \( \ln(x) \), it asks: "What power must \( e \) be raised to, in order to obtain \( x \)?" Like other logarithmic functions, the natural logarithm can simplify complex multiplicative relationships into manageable additive ones.
This concept is useful in various fields, from finance to physics, where understanding rates of change is crucial.
Exploring Mathematical Approximation
Mathematical approximation helps us find values close to exact solutions but can be computed easily without complex calculations. In the context of natural logarithms, approximations are essential for estimating values without a calculator.
One of the most straightforward approximations for the natural logarithm is that \( \ln(1+x) \approx x \), when \( x \) is close to 0. This approximation is derived from the Taylor series expansion for \( \ln(1+x) \) but simplified for small values of \( x \).
Such approximations are common in problems where precision isn't as critical, and a quick estimation suffices. They are also crucial in settings where computational resources are limited.
Defining Logarithmic Function
A logarithmic function is a type of inverse function that undoes the exponential operation. The general form \( y = \log_b(x) \) indicates that \( b \) raised to the power of \( y \) equals \( x \).
Each base, \( b \), provides a different family of logarithmic functions. When \( b = e \), the function becomes the natural logarithm. Logarithmic functions are used to model phenomena that grow exponentially, translating multiplicative processes into simpler, linear ones.
For instance, if you double an amount continuously over time, the logarithmic function tells you how long it takes in terms of the logarithm of the multiplier.
Effective Problem-Solving Steps
Solving mathematical problems often involves structured steps. When we're asked to estimate \( \ln(0.993) \) without a calculator, a step-by-step method can simplify the process.
First, express the number close to 1 to take advantage of simple approximations. In this case, rewrite \( 0.993 \) as \( 1 - 0.007 \). Next, apply the rough approximation \( \ln(1+x) \approx x \), which understates the difficulty of calculating logarithms directly. This yields \( \ln(0.993) \approx -0.007 \).
The process reiterates problem-solving principles: breaking down complex problems into easier parts, using approximations effectively, and achieving a solid estimate with minimal computation.

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