Problem 12 - Precalculus Mathematics for Calculus

Question

Express the equation in exponential form. (a) \(\ln (x+1)=2\) (b) \(\ln (x-1)=4\)

Step-by-Step Solution

Answer

(a) \( x = e^2 - 1 \), (b) \( x = e^4 + 1 \).

Step 1: Understand the Natural Logarithm

The natural logarithm function, denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \approx 2.71828 \). This means \( \ln(a) = b \) can be rewritten in its exponential form as \( e^b = a \).

Step 2: Rewrite Equation (a) in Exponential Form

Given the equation \( \ln(x+1) = 2 \), use the property of natural logarithms: \( \ln(a) = b \) implies \( e^b = a \). Therefore, \( e^2 = x + 1 \).

Step 3: Solve for x in Equation (a)

Isolate \( x \) by subtracting 1 from both sides: \( e^2 - 1 = x \). Hence, in exponential form, \( x = e^2 - 1 \).

Step 4: Rewrite Equation (b) in Exponential Form

Given the equation \( \ln(x-1) = 4 \), apply the same property: \( e^b = a \). Thus, \( e^4 = x - 1 \).

Step 5: Solve for x in Equation (b)

Isolate \( x \) by adding 1 to both sides: \( x = e^4 + 1 \).

Key Concepts

Natural Logarithm

The natural logarithm is a fundamental concept in mathematics, often symbolized by \( \ln \). It is the logarithm with the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828.
Understanding the natural logarithm involves recognizing its unique base, \( e \), which is intrinsic to growth and decay problems in calculus and natural processes. When you see a natural logarithm, such as \( \ln(a) = b \), it means that \( e \) raised to the power of \( b \) equals \( a \).
Benefits of natural logarithms include:

• Simplifying expressions involving exponential growth or decay.
• Providing solutions to problems involving continuous growth.
• Offering a way to transform multiplication operations into additions.

Being comfortable with working from logarithmic and exponential forms is key to successfully navigating through related mathematical problems.

Exponential Equations

Exponential equations are mathematical expressions where variables appear as exponents. They play a crucial role in many real-world contexts, such as compound interest, population growth, and radioactive decay.
In any exponential equation, the expression follows the format \( a^x = b \), where \( a \) is a constant, \( x \) is the exponent, and \( b \) is the result of the expression. Solving these equations often involves using logarithms to isolate the variable.
To illustrate, consider the equation \( e^x = y \). Here, \( e \) is the base, and by taking the natural logarithm of both sides, you can solve for \( x \) to find that \( x = \ln(y) \).
Understanding exponential equations includes:

• Identifying the base and variable.
• Utilizing logarithmic conversion to solve for the exponent.
• Recognizing how exponential growth or decay affects the equation.

Mastering these equations enables one to address complex mathematical models with efficiency.

Logarithmic to Exponential Conversion

Converting between logarithmic and exponential forms is an essential skill in algebra and precalculus. It helps in simplifying problems and solving exponential equations.
The conversion follows a straightforward principle: if you have a logarithmic equation such as \( \ln(a) = b \), it can be rewritten in exponential form as \( e^b = a \). This transformation arises from the definition of logarithms as the inverse of exponentiation.
Steps for conversion:

• Identify the base of the natural logarithm, which is always \( e \).
• Translate the logarithmic equation into its equivalent exponential sentence.
• Use the exponential form for solving the equation if needed.

By mastering this conversion, you gain the ability to seamlessly navigate between expressions and understand the relationships between variables within equations.