Problem 37 - Algebra 2

Question

Mental Math Simplify each expression. \(\frac{\ln e^{2}}{2}\)

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Recognize the base of the logarithm

The base of the logarithm in this expression, \(\ln\), is the natural logarithm base, also known as \(e\). This is the same base as the base of the exponential function in the numerator, \(e^{2}\). This will simplify the expression.

Step 2: Use laws of logarithms

Since the base of the logarithm and the base of the exponent are the same, it simplifies to just the value of the exponent. Therefore, \(\ln e^{2}\) simplifies to 2.

Step 3: Simplify the expression

The expression becomes \(\frac{2}{2}\), which simplifies further into 1.

Key Concepts

Natural Logarithm

The natural logarithm, often denoted by "ln", is a special type of logarithm. Its base is the irrational number denoted as "e.", where approximately, \( e \approx 2.71828 \). The natural logarithm has unique properties that make it particularly useful in calculus and real-world models involving growth and decay.
A key feature of the natural logarithm is its relation with the exponential function, meaning it helps delve into situations involving exponential growth or decay. In simpler terms, if you have a number expressed as \( e^x \), the natural log \( \ln(e^x) \) equals to the exponent "x". This is because of the fundamental property of logarithms that \( \ln(e) = 1 \).
The natural log also has some relationships with continuous compounding and interest formulas, which you might encounter in finance. But remember, when you see \( \ln \), always connect it with \( e \), its natural pair!

Exponential Function

An exponential function is defined by the mathematical function expressed in the form \( f(x) = e^x \). Here, "e" is the base, and "x" is an exponent. Exponential functions are crucial in mathematics due to their constant growth rates and applications in science and engineering.
A fundamental characteristic of exponential functions is their ability to model scenarios where the rate of change is proportional to the current value, such as population growth or radioactive decay.

• Exponential growth: For values of "x" greater than zero, \( e^x \) describes an increasing curve.
• Exponential decay: Conversely, for negative "x" values, \( e^{-x} \) describes a decreasing curve.

Connecting exponential functions and natural logarithms is key. For example, in the original exercise \( \ln(e^2) \): the natural log reverses the effect of the exponent, unraveling "2" as the solution due to the natural relationship between these operations.

Logarithm Laws

Logarithm laws are essential tools that help simplify and manipulate logarithmic expressions. When dealing with logarithms, these rules allow us to break down complex power expressions into more manageable forms.
There are some essential laws to keep in mind:

• Product law: \( \log_b(MN) = \log_b(M) + \log_b(N) \)
• Quotient law: \( \log_b\left( \frac{M}{N} \right) = \log_b(M) - \log_b(N) \)
• Power law: \( \log_b(M^k) = k \cdot \log_b(M) \)
• Change of base formula: \( \log_b(M) = \frac{\log_k(M)}{\log_k(b)} \)

In the original exercise, we specifically use the power law. With \( \ln(e^2) \), we can directly use the power law to see \( \ln(e^2) = 2\cdot \ln(e) \), and since \( \ln(e) = 1 \), this simplifies to "2". Understanding these laws helps navigate through more advanced logarithmic problems with ease.