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Chapter 3: Problem 93

Evaluate or simplify each expression without using a calculator. $$e^{\ln 125}$$

Short Answer

Expert verified
So, \(e^{\ln 125} = 125\)

Step by step solution

01

Understanding the problem

The expression given is \(e^{\ln 125}\). Follow the order of operations (PEMDAS/BODMAS), where powers or exponents are calculated before logarithms.
02

Executing the natural logarithm and exponential functions

Since the natural logarithm (\ln) and base e exponentiation (or e raised to some power) are inverse functions, they cancel each other out, effectively leaving what was inside the \ln function. Therefore, \(e^{\ln 125}\) simplifies to 125.

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Key Concepts

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

Understanding Natural Logarithms
Natural logarithms are a specific type of logarithm that use the base \(e\), where \(e\) is an irrational number approximately equal to 2.71828. In mathematical terms, the natural logarithm of a number \(x\), denoted as \(\ln x\), represents the exponent you need to raise \(e\) to in order to get the number \(x\).

For example, if \(\ln x = 2\), it means that \(e^2 = x\). Natural logarithms are widely used in calculus and exponential growth problems, such as interest calculations and population growth models, because they simplify differentiation and integration processes.

Important points to remember about natural logarithms:
  • \(\ln e = 1\) because \(e^1 = e\).
  • The natural logarithm of 1 is 0, i.e., \(\ln 1 = 0\), because any number raised to the power of 0 is 1.
  • \(\ln\) and exponential functions using base \(e\) are inverse operations, meaning they cancel each other out. For example, if you have \(e^{\ln x}\), it simplifies to \(x\).
Inverse Functions and Their Role
An inverse function essentially reverses the effect of a function, taking you back to the original value before the function was applied. When we talk about an exponential function like \(e^x\) and its corresponding inverse, the natural logarithm \(\ln x\), they undo each other鈥檚 effects.

This means if you apply one, and then the other, you return to your starting point.

Why inverse functions are important:
  • They allow us to solve equations that involve exponential growth or decay.
  • They are crucial in calculus for solving differential equations and integration problems.
  • Applications often appear in real-world scenarios such as physics, finance, and biology.

As demonstrated in the problem \(e^{\ln 125}\), since \(e\) and \(\ln\) are inverses, they cancel out, simplifying the expression directly to 125.
Simplifying Expressions with Ease
Simplifying expressions involves making them easier to read or solve without changing their value. It's about reducing them to their simplest form. For the expression \(e^{\ln 125}\), simplifying involves recognizing that the natural logarithm \(\ln\) and the base \(e\) exponential are inverse operations.

This relationship allows them to effectively "cancel" each other out, which simplifies expressions significantly.

Steps to simplify expressions involving inverse operations:
  • Identify parts of the expression that are inverse functions, such as \(e\) and \(\ln\).
  • Apply the property that inverse functions "undo" each other.
  • Simplify the expression by removing these inverse parts, in our example, directly turning \(e^{\ln 125}\) into 125.

This process not only saves time but also reduces complexity, making it far easier to work with exponential and logarithmic problems in both mathematics and the sciences.

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Most popular questions from this chapter

Use Newton's Law of Cooling, \(T=C+\left(T_{0}-C\right) e^{k t},\) to solve this exercise. At 9: 00 A.M., a coroner arrived at the home of a person who had died. The temperature of the room was \(70^{\circ} \mathrm{F}\), and at the time of death the person had a body temperature of \(98.6^{\circ} \mathrm{F} .\) The coroner took the body's temperature at 9: 30 A.M., at which time it was \(85.6^{\circ} \mathrm{F},\) and again at 10: 00 A.M., when it was \(82.7^{\circ} \mathrm{F} .\) At what time did the person die?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The functions \(f(x)=\left(\frac{1}{3}\right)^{x}\) and \(g(x)=3^{-x}\) have the same graph.

The formula \(S=C(1+r)^{t}\) models inflation, where \(C=\) the value today, \(r=\)the annual inflation rate, and \(S=\)the inflated value t years from now. Use this formula to solve. Round answers to the nearest dollar. If the inflation rate is \(6 \%,\) how much will a house now worth \(\$ 465,000\) be worth in 10 years?

Rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places. $$y=4.5(0.6)^{x}$$

Suppose that a population that is growing exponentially increases from 800,000 people in 2010 to 1,000,000 people in \(2013 .\) Without showing the details, describe how to obtain the exponential growth function that models the data.
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