Problem 97 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$\ln e^{3}-\ln e^{7}$$.

Step-by-Step Solution

Answer

The exact value of given expression \( \ln e^{3} - \ln e^{7} \) is -4

Step 1: Apply Logarithmic Identity \(ln(e^{x})=x\)

By using the logarithm identity \( \ln e^{x} = x \), we get \( \ln e^{3} - \ln e^{7} = 3 - 7 \)

Step 2: Perform Subtraction

Subtracting 7 from 3, we get \( 3 - 7 = -4 \)

Key Concepts

Natural Logarithm

The natural logarithm, often represented as \(\ln\), is a logarithm whose base is the mathematical constant \( e \). The constant \( e \) is approximately equal to 2.71828, and it appears in many areas of mathematics, especially in calculus and complex analysis. The natural logarithm of a number \( x \), denoted \( \ln(x) \), solves the equation \( e^y = x \). In simpler terms, \( \ln(x) \) answers the question: to what power must \( e \) be raised to yield \( x \)?

The natural logarithm has several useful properties, such as:

• \( \ln(1) = 0 \) because \( e^0 = 1 \).
• \( \ln(e) = 1 \) because \( e^1 = e \).
• For any positive numbers \( a \) and \( b \), \( \ln(ab) = \ln(a) + \ln(b) \).
• \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).

Understanding these properties helps simplify logarithmic expressions, as seen in the original exercise. Using the fact that \( \ln(e^x) = x \), you can quickly evaluate the expression \( \ln e^3 - \ln e^7 \) as \( 3 - 7 \).

Exponential Function

The exponential function \( f(x) = e^x \) is a fundamental mathematical function where \( e \) is the base and \( x \) is the exponent. It grows very quickly and has the unique property that its rate of change is proportional to its current value. This makes it incredibly useful in modeling growth and decay processes, such as population growth or radioactive decay.

Here are some important characteristics of the exponential function:

• The function \( e^x \) is always positive for any real number \( x \).
• The derivative of the exponential function is itself: \( \frac{d}{dx} e^x = e^x \).
• Its inverse function is the natural logarithm \( \ln(x) \).
• It has horizontal asymptotes at \( y = 0 \) as \( x \to -\infty \).

The interplay between the exponential function and the natural logarithm is central to solving many mathematical problems, including those involving exponential growth and decay.

In the context of the original exercise, the property \( \ln(e^x) = x \) is directly used to simplify the expression \( \ln e^3 - \ln e^7 \). This shows the important relationship between logarithms and exponents.

Logarithmic Identities

Logarithmic identities are mathematical tools that simplify logarithmic expressions and solve logarithmic equations. They stem from the fundamental properties of logarithms and are essential for making complex calculations more manageable.

Here are some key logarithmic identities:

• \( \log_b(xy) = \log_b(x) + \log_b(y) \)
• \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \)
• \( \log_b(x^y) = y \log_b(x) \)
• \( \log_b(b) = 1 \)
• \( \log_b(1) = 0 \)

These identities are particularly useful when dealing with expressions involving multiplication, division, or powers. For the case of natural logarithms, which base is \( e \), the identity \( \ln(e^x) = x \) is invaluable for simplifying expressions, as demonstrated in our given exercise.

In the original solution, the logarithmic identity \( \ln(e^x) = x \) helps to transform the expression \( \ln e^3 - \ln e^7 \) into \( 3 - 7 \), showcasing how identities can simplify and solve logarithmic equations efficiently.