Problem 40 - College Algebra

Question

For the following exercises, use the definition of common and natural logarithms to simplify. $$ \ln \left(e^{-5.03}\right) $$

Step-by-Step Solution

Answer

\(-5.03\)

Step 1: Identify the relationship

Recognize the property of logarithms which states that if you have the natural logarithm of an exponent with base \(e\), such as \(\ln(e^x)\), it simplifies directly to just \(x\).

Step 2: Apply the property

Given the expression \(\ln(e^{-5.03})\), apply the property from Step 1, where \(x = -5.03\). This simplifies to \(-5.03\) directly.

Step 3: Simplify the expression

The natural logarithm function and the exponential function are inverses. Therefore, simplifying \(\ln(e^{-5.03})\) gives us \(-5.03\).

Key Concepts

Common Logarithms

Understanding common logarithms is crucial in solving logarithmic expressions. Common logarithms are logarithms with base 10. When you see "log" without any base specified, it usually implies a common logarithm. This representation follows because the number 10 is a standard base used in our decimal numbering system. Common logarithms are instrumental in scientific measurements and calculations, helping us simplify and manage large numbers more effectively. Here are some quick points about common logarithms:

• If you have a number like 1000 and want to find its common logarithm, you are essentially asking what power you must raise 10 to get 1000. In this example, it would be 3, because 10³ = 1000.
• They are denoted as "log" with no base, implying a base of 10 by default.

These are different from natural logarithms, which use Euler's number, \(e\), as the base. Despite their differences, the techniques to solve them share many similarities, mostly due to the properties of logarithms.

Logarithmic Properties

Several properties make solving logarithmic equations easier. These come from the way logarithms function as exponents. Here are some essential properties:

• Power Rule: \(\log_b(x^y) = y \cdot \log_b(x)\). This means that when you take the logarithm of a power, you can bring down the exponent as a coefficient.
• Product Rule: \(\log_b(xy) = \log_b(x) + \log_b(y)\). This enables you to split a logarithm of a product into two separate logarithms.
• Quotient Rule: \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\). When dealing with division inside a logarithm, it can be separated into the subtraction of two logs.

Understanding these properties is key to simplifying complex logarithmic expressions. For example, in the original exercise, the property \(\ln(e^x)=x\) was used to simplify \(\ln(e^{-5.03})\) to \(-5.03\). This relies on the fact that logarithms and exponentials are inverse operations, and thus simplify directly without calculation.

Exponential Functions

Exponential functions are rapidly growing mathematical functions represented by the form \(f(x) = a \cdot e^{bx}\), where \(e\) is Euler's number, approximately 2.71828. These functions are powerful in modeling growth processes in real life such as population growth, radioactive decay, and many natural processes. The inverse of an exponential function is a logarithm. For example, if you have an exponential function \(e^x\), the inverse operation is \(\ln(e^x) = x\), showcasing their relationship.

• Exponential growth happens when \(b\) is positive, resulting in a graph that rises steeply.
• Exponential decay occurs when \(b\) is negative, leading to a graph that falls rapidly as it moves.

Understanding exponential functions and their inverse relationship with logarithms allows us to handle and transform complex expressions seamlessly, just like simplifying \(\ln(e^{-5.03})\) using properties from both these mathematical concepts.