Problem 12 - Algebra and Trigonometry

Question

Express the equation in exponential form. $$ \ln (x+1)=2 \quad \text { (b) } \ln (x-1)=4 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Natural Logarithm

The given equations involve the natural logarithm function, which is denoted as \( \ln \). The natural logarithm of a number is the power to which \( e \) (the base of the natural logarithm, approximately 2.718) must be raised to produce that number. Thus, \( \ln(a) = b \) can be rewritten as \( e^b = a \).

Step 2: Convert the First Equation to Exponential Form

We start with the first equation \( \ln(x+1) = 2 \). According to the property of logarithms, this equation can be rewritten in its exponential form as \( e^2 = x + 1 \).

Step 3: Solve for x in the First Equation

To find \( x \), we rearrange the equation \( e^2 = x + 1 \) to solve for \( x \). Subtract 1 from both sides to get \( x = e^2 - 1 \).

Step 4: Convert the Second Equation to Exponential Form

The second equation \( \ln(x-1) = 4 \) can also be converted to exponential form. It can be rewritten as \( e^4 = x - 1 \).

Step 5: Solve for x in the Second Equation

To solve for \( x \) in the second equation, rearrange \( e^4 = x - 1 \) to obtain \( x = e^4 + 1 \).

Key Concepts

Natural Logarithm

The natural logarithm is a special type of logarithm that has the base of Euler's number, denoted as \( e \), which is approximately 2.718. It's widely used in mathematics to represent continuous growth, such as in natural phenomena or compound interest.

• Natural logarithms are noted by \( \ln \), unlike common logarithms which use base 10 and are denoted by \( \log \).
• The natural logarithm of a number tells us the exponent that \( e \) must be raised to achieve that number.

Take, for example, \( \ln(a) = b \). Here, \( b \) is the power that \( e \) must be raised to get \( a \). This relationship can be simplified into its exponential form as \( e^b = a \). This property is crucial for transforming logarithmic equations into exponential equations for easier manipulation and solving.

Logarithms

Logarithms are the operations that find out "how many times a base number is multiplied to produce a certain number". The logarithm base is often 10 in common logarithms, or \( e \) in natural logarithms.

• Logarithmic functions are the inverse of exponential functions. This means that if \( b^y = x \), then \( \log_b(x) = y \).
• Understanding this inverse relationship helps to solve logarithmic equations by converting them into exponential form.

Converting equations like \( \ln(x+1) = 2 \) into exponential form using the knowledge of logarithms makes solving such equations manageable. By rewriting \( \ln(x+1) = 2 \) as \( e^2 = x+1 \), we can more readily solve for \( x \). Similarly, \( \ln(x-1) = 4 \) converts to \( e^4 = x-1 \), leading directly to the solution.

Exponential Equations

Exponential equations are algebraic expressions where variables appear as exponents. They can be challenging due to their non-linear nature. However, converting logarithmic equations to exponential form simplifies finding solutions for the variable.Here are some key points:

• Start by understanding the properties of exponential functions. They grow rapidly compared to polynomial or linear functions.
• The general form of an exponential equation is \( b^x = y \). For natural logarithms, \( e^x = y \).

By converting \( \ln(x+1)=2 \) to \( e^2 = x+1 \), solving for \( x \) becomes straightforward: subtract 1 from both sides. Similarly, turning \( \ln(x-1)=4 \) into \( e^4 = x-1 \) helps isolate \( x \) by adding 1 to both sides. With this approach, exponential equations become manageable, tapping into the essence of their growth behavior.