Problem 18 - Precalculus

Question

Apply the properties of logarithms to simplify each expression. Do not use a calculator. $$e^{-2 \ln 10}$$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{1}{100} \).

Step 1: Understand the Problem

We need to simplify the expression \( e^{-2 \ln 10} \) using properties of logarithms and exponents.

Step 2: Use the Power Rule of Logarithms

Recognize that the expression \( e^{-2 \ln 10} \) can be rewritten using the power rule of logarithms as \( e^{\ln (10^{-2})} \). This uses the property \( a \cdot \ln b = \ln(b^a) \).

Step 3: Apply the Exponential and Logarithm Inverse Property

The exponential function and the natural logarithm are inverse functions, meaning \( e^{\ln x} = x \). Therefore, \( e^{\ln (10^{-2})} = 10^{-2} \).

Step 4: Simplify the Expression

Now calculate \( 10^{-2} \). This is equal to \( \frac{1}{10^2} = \frac{1}{100} \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. They are a key concept in algebra and calculus due to their ability to model growth or decay processes, such as population growth or radioactive decay. The general form of an exponential function is given by \[ f(x) = a \cdot b^x \]where:

• \(a\) is the initial value or coefficient,
• \(b\) is the base of the exponential function, which must be a positive real number, and
• \(x\) is the exponent or power.

Some notable properties of exponential functions include:

• If the base \(b > 1\), the function represents exponential growth.
• If \(0 < b < 1\), the function represents exponential decay.
• Exponential functions have a horizontal asymptote, usually the x-axis (\(y = 0\)).

In the context of the problem, the exponential function involved is \(e\), which is the base of natural logarithms. This makes working with natural logarithms much simpler as they are inverses.

Natural Logarithms

Natural logarithms are the logarithms with the base of \(e\), where \(e\) is approximately equal to 2.71828. They are denoted as \(\ln \). Natural logarithms are crucial in calculus and the natural sciences because they arise in many natural phenomena and equations. The natural logarithm of a number \(x\) is the power to which \(e\) must be raised to get \(x\). This means\[ \ln(e^x) = x \quad \text{and} \quad e^{\ln x} = x \]These properties particularly highlight the inverse relationship between natural logarithms and the exponential function.Here's why the inverse property is helpful:

• Transformations like \(e^{\ln (10^{-2})}\) simplify directly to \(10^{-2}\).
• This results from the exponentiation and the logarithmic process canceling each other.

Understanding this inverse relationship is key to simplifying expressions involving natural logarithms and exponential functions.

Power Rule of Logarithms

The power rule of logarithms is an essential property used to simplify and solve logarithmic expressions. It states:\[ a \cdot \ln b = \ln(b^a) \]This property enables the exponent (power) to be moved from a coefficient in front of a logarithm to within the logarithmic argument as an exponent. With this rule, complicated expressions become easier to handle because they allow us to rewrite and simplify the problem.For example, consider the expression \( -2 \cdot \ln 10 \). Using the power rule, this can be rewritten as:\[ \ln(10^{-2}) \]This simplification was a key step in solving the problem at hand. By recognizing and applying this rule, complex expressions involving logarithms are made more straightforward, thereby aiding further simplification with the exponential functions and their inverses.