Problem 499 - Precalculus

Question

Rewrite \(\ln (s)=t\) as an equivalent exponential equation.

Step-by-Step Solution

Answer

The equivalent exponential equation is \( e^t = s \).

Step 1: Understanding the Logarithmic Equation

The given equation is \( \ln(s) = t \) where \( \ln \) represents the natural logarithm, which has a base of \( e \), the mathematical constant approximately equal to 2.71828.

Step 2: Setting Up the Equivalent Exponential Equation

In general, a logarithmic equation \( \ln(a) = b \) can be rewritten in exponential form as \( e^b = a \). Here, \( a \) is the result when \( e \) is raised to the power of \( b \).

Step 3: Applying the Rule to the Given Equation

By applying the rule from the previous step, we rewrite \( \ln(s) = t \) as \( e^t = s \). This is derived by recognizing that if \( t \) is the power to which \( e \) must be raised to get \( s \), then \( e^t \) must equal \( s \).

Key Concepts

Natural Logarithms

The concept of natural logarithms is central to understanding exponential functions. A natural logarithm, denoted as \( \ln \), uses the mathematical constant \( e \) as its base.

• \( e \) is approximately equal to 2.71828 and is known as Euler's number.
• Natural logarithms are used to determine how much time it takes for an amount to grow to a certain level with continuous compounding.
• They are widely used in natural sciences, economics, and fields that involve exponential growth or decay.

The natural logarithm function \( \ln(s) \) represents the power to which \( e \) must be raised to result in \( s \). Hence, understanding \( \ln \) involves recognizing how it relates to exponential expressions. This understanding helps simplify and solve equations where exponential processes are involved.

Logarithmic Equations

Logarithmic equations involve solving for unknowns within a logarithmic expression. In the example \( \ln(s) = t \), we see a straightforward logarithmic equation. To convert this to an exponential form:- We use the general formula \( \ln(a) = b \), which represents that \( e^b = a \).- Here, \( e^t = s \) is the exponential equivalent of our logarithmic equation.

• Solving logarithmic equations often involves rewriting the logarithm in its exponential form.
• This transformation helps because exponential forms can be easier to work with or compare.
• Using properties of logarithms and exponents can simplify complex equations or inequalities, making them easier to solve.

Converting between logarithmic and exponential forms is a key skill in algebra, allowing for deeper understanding and more versatile problem-solving methods.

Base e

Base \( e \) is a fundamental part of both exponential and logarithmic functions. Often encountered in equations related to growth and decay, \( e \) is a transcendental number that provides a natural base for logarithms.

• The value of \( e \) is approximately 2.71828, but it appears in complex formulas across sciences and finance.
• Exponential functions with base \( e \) describe continuous growth, seen in populations, compound interest, and radioactive decay.

Understanding \( e \) as a base:- When we look at equations like \( e^t = s \), we grasp how \( e \) facilitates transformations between growth stages.- Since \( e \) leads to smooth, continuous growth, it's known as the "natural" base, distinguishing it from other bases like 10.Grasping this concept allows for applying natural logarithms and equations across various real-world applications, from engineering problems to complex financial forecasting.