Problem 15
Question
Evaluate each expression to four decimal places using a calculator. $$e^{-2.5}$$
Step-by-Step Solution
Verified Answer
After evaluating the given expression to four decimal places, the result should be 0.0821.
1Step 1: Understand the nature of 'e'
The 'e' in the expression represents the base of the natural logarithms, which is a mathematical constant approximately equal to 2.71828.
2Step 2: Understand the negative exponent
The negative in the exponent means we take the reciprocal of the base. So, if we have \(a^{-n}\), it is the same as \(\frac{1}{a^{n}}\). In this case, \(e^{-2.5}\) can also be written as \(\frac{1}{e^{2.5}}\).
3Step 3: Evaluate using a calculator
To perform the above operation, use a calculator that has the 'e' (Euler's number) function. Enter 'e', then '^', then '(-2.5)'. Alternatively, enter '1', divide it by 'e' raised to '2.5'. Make sure to round the result to four decimal places as the question instructs.
Key Concepts
Base of Natural LogarithmsNegative ExponentsNumerical Calculation
Base of Natural Logarithms
When we talk about the base of natural logarithms, we're referring to the irrational number 'e', also known as Euler's number. This constant is an integral part of exponential and logarithmic functions, particular in continuous growth and decay models. It is approximately equal to 2.71828.
Why is 'e' so important? It arises naturally in the process of solving problems involving growth rates and can be found in many areas of mathematics, including calculus and complex numbers. In simpler terms, think of 'e' as the base rate of growth shared by all continually growing processes. For instance, when you place money into an account with continuous compounding interest, the value of 'e' will play a role in calculating the compounded interest over time.
Understanding 'e' is crucial not just for evaluating exponential expressions but for grasping the natural growth patterns that it represents, which are foundations of many real-world phenomena from population growth to radioactive decay.
Why is 'e' so important? It arises naturally in the process of solving problems involving growth rates and can be found in many areas of mathematics, including calculus and complex numbers. In simpler terms, think of 'e' as the base rate of growth shared by all continually growing processes. For instance, when you place money into an account with continuous compounding interest, the value of 'e' will play a role in calculating the compounded interest over time.
Understanding 'e' is crucial not just for evaluating exponential expressions but for grasping the natural growth patterns that it represents, which are foundations of many real-world phenomena from population growth to radioactive decay.
Negative Exponents
Exponents are not always positive; they can be negative too. When you encounter a negative exponent, it signifies that you should take the reciprocal (or inverse) of the base raised to the positive version of that exponent.
For example, a base 'a' with an exponent of '-n' is the same as \(\frac{1}{a^n}\). This is essential to understand because it transforms the expression into a fraction, which fundamentally changes how it's calculated. In the context of the exercise with \(e^{-2.5}\), expressing it as \(\frac{1}{e^{2.5}}\) reveals that we're actually dealing with a fraction of 1 divided by a certain growth factor.
The concept of negative exponents becomes especially relevant when dealing with decay processes or phenomena that decrease over time. In our monetary example, if a bank used a negative interest rate, the formula to calculate the decay in value would use a negative exponent.
For example, a base 'a' with an exponent of '-n' is the same as \(\frac{1}{a^n}\). This is essential to understand because it transforms the expression into a fraction, which fundamentally changes how it's calculated. In the context of the exercise with \(e^{-2.5}\), expressing it as \(\frac{1}{e^{2.5}}\) reveals that we're actually dealing with a fraction of 1 divided by a certain growth factor.
The concept of negative exponents becomes especially relevant when dealing with decay processes or phenomena that decrease over time. In our monetary example, if a bank used a negative interest rate, the formula to calculate the decay in value would use a negative exponent.
Numerical Calculation
The field of numerical calculation involves computing exact or approximate results, often by using digital tools such as calculators. For students, mastering the use of calculators to perform operations with constants like 'e' is an invaluable skill.
In our given expression \(e^{-2.5}\), to calculate this value numerically, a calculator with an 'e' function is essential. Here’s how you can do it: Input 'e', press the exponent key, typically labeled as '^', then input '-2.5', and hit the equals sign. If your calculator lacks an 'e' function, use the reciprocal method mentioned earlier. This means you would enter \(e^{2.5}\), get the result, and then divide 1 by that result to find your answer for \(e^{-2.5}\).
Rounding to four decimal places, as instructed by the exercise, is the final step in ensuring your answer is accurate and clear. This precision is often required in scientific and engineering practices, where exact values are paramount. Keep in mind that regardless of whether the operations seem simple, attention to detail is key in numerical calculations to avoid errors.
In our given expression \(e^{-2.5}\), to calculate this value numerically, a calculator with an 'e' function is essential. Here’s how you can do it: Input 'e', press the exponent key, typically labeled as '^', then input '-2.5', and hit the equals sign. If your calculator lacks an 'e' function, use the reciprocal method mentioned earlier. This means you would enter \(e^{2.5}\), get the result, and then divide 1 by that result to find your answer for \(e^{-2.5}\).
Rounding to four decimal places, as instructed by the exercise, is the final step in ensuring your answer is accurate and clear. This precision is often required in scientific and engineering practices, where exact values are paramount. Keep in mind that regardless of whether the operations seem simple, attention to detail is key in numerical calculations to avoid errors.
Other exercises in this chapter
Problem 15
Evaluate each expression without using a calculator. $$\log 10,000$$
View solution Problem 15
Verify that the given functions are inverses of each other. $$f(x)=x^{3}+2 ; g(x)=\sqrt[3]{x-2}$$
View solution Problem 16
Use \(f(x)=\frac{10}{1+2 e^{-0.3 x}}\) Evaluate \(f(1)\).
View solution Problem 16
Solve the exponential equation. Round to three decimal places, when needed. $$6\left(0.9^{x}\right)=7$$
View solution