Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the logarithmic equation in exponential form. For example, the exponential form of \(\ln 5=1.6094\). . . is \(e^{1.6094 \cdots}=5.\) $$\ln e^{3}=3$$

Step-by-Step Solution

Answer

The exponential form of the given logarithmic equation \(\ln e^{3} = 3\) is \(e^{3} = e^{3}\).

Step 1: Understand the logarithmic properties

Recall that ln is the natural logarithm, which is the logarithm to the base e, where e is an irrational and transcendental number approximately equal to 2.71828182846. Additionally, the exponential function and the natural logarithm are inversely related, and they can be transformed into one another. From this, we know that if the equation \(ln(e^{x}) = x\) holds for any real number x, we can rewrite it in its exponential form as \(e^{x} = e^{x}\).

Step 2: Transform the equation

Taking \(\ln e^{3} = 3\), we can now use our knowledge to rewrite it in its exponential form. Applying the property, we get: \(e^{3} = e^{3}\).

Key Concepts

Exponential Form

The exponential form of an equation is a way of expressing the relationship between logarithms and exponents. To convert a logarithmic equation to its exponential form, we focus on the base of the logarithm. The base, when raised to the result of the logarithm, will give us the initial value inside the log function.

For natural logarithms, the base is the constant \( e \), approximately equal to 2.718. When you have a natural logarithm like \( \ln 5 = 1.6094 \), this indicates that the constant \( e \) raised to 1.6094 equals 5.

In practice:

• Identify the base — for natural logarithms, it's always \( e \).
• The exponent is the value to which the base is raised, giving the inside term of the logarithm.
• Write it in the form \( e^{\text{logarithm's value}} = \text{result inside ln} \).

Understanding exponential forms helps in recognizing relationships between logarithmic functions and their inverses.

Natural Logarithm

The natural logarithm, often denoted as \( \ln \), is a logarithm with base \( e \). It plays a crucial role in calculus and various mathematical applications due to its natural occurrence in mathematical settings.

Here's what makes the natural logarithm special:

• The number \( e \) (approximately 2.718) is a natural base that appears frequently across mathematics, especially in growth and decay processes.
• \( \ln \) is the inverse of the exponential function \( e^{x} \). This relationship facilitates switching between logarithmic and exponential equations.

For example, with \( \ln e^3 = 3 \), the natural logarithm \( \ln \) simplifies to reveal the exponent. Therefore, it states that the exponent by which \( e \) must be raised to yield \( e^3 \) is 3. This ability to convert to simpler forms aids in handling complex equations.

Natural logarithms help simplify when solving for exponents in calculus or working with specific bases, offering a streamlined approach to otherwise challenging processes.

Logarithmic Properties

Logarithmic properties are fundamental rules that describe how logarithms operate and relate to one another. These properties allow us to manipulate and solve logarithmic equations like \( \ln e^3 = 3 \).

Here are a few core properties to remember:

• Product Property: \( \ln(ab) = \ln(a) + \ln(b) \). This shows how multiplication inside a logarithm can be expanded to addition outside.
• Quotient Property: \( \ln(a/b) = \ln(a) - \ln(b) \). This property allows us to separate division operations inside the log.
• Power Property: \( \ln(a^b) = b \cdot \ln(a) \). This translates a power inside into a multiplication outside, pivotal in simplifying equations.
• Base Change: Converting bases using the formula \( \log_b(a) = \frac{\ln(a)}{\ln(b)} \) when needed.

These properties are essential for simplifying and solving logarithmic problems. They make it possible to transform complex expressions into manageable forms, making equations easier to analyze and solve. By mastering these, you can navigate a wide range of problems with confidence.