Problem 94 - Algebra and Trigonometry

Question

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$ \ln (x+1)=\ln x+\ln 1 $$

Step-by-Step Solution

Answer

The given equation is false. The correct equation is \( \ln (x . 1) = \ln x + \ln 1\).

Step 1: Identify each part of the equation

The equation given is \( \ln (x+1)=\ln x+\ln 1 \). Here, \(\ln (x + 1)\) is the natural logarithm of \(x + 1\), \(\ln x\) is the natural logarithm of \(x\), and \(\ln 1\) is the natural logarithm of \(1\).

Step 2: Evaluate the known logarithm

The natural logarithm of \(1\) is \(0\), since any number raised to the power \(0\) equals \(1\). So, \(\ln 1\) can be replaced by \(0\). Thus, the equation becomes: \( \ln (x+1)=\ln x+0\), i.e., \(\ln (x+1)=\ln x\).

Step 3: Compare both sides of the equation

For the equation, \(\ln (x + 1) = \ln x\), to be true, the arguments of the logarithms, i.e., \(x + 1\) and \(x\), should be equal. However, \(x + 1\) is not equal to \(x\). Hence, the given equation is false.

Step 4: Correct the equation

For the original statement to become correct, both sides of the equation must be equal. The correct equation will thus be \( \ln (x . 1) = \ln x + \ln 1\), as the logarithm of a product is the sum of the logarithms of the factors.

Key Concepts

Natural Logarithm

The natural logarithm, often notated as \( \ln \), is a special type of logarithm that uses the constant base \( e \), where \( e \approx 2.71828 \). It is widely used in mathematics due to its natural occurrence in various exponential growth scenarios, such as population dynamics and compound interest.

• It is crucial to remember that \( \ln(e^x) = x \), and \( e^{\ln x} = x \), as these relationships emphasize the inverse nature of exponential and logarithmic functions.
• Another important property is that the natural logarithm of 1, \( \ln 1 \), always equals 0. This is because any number raised to the power of 0 is 1.

These qualities of the natural logarithm make it powerful for simplifying and solving equations where exponential and logarithmic expressions are involved. When solving logarithmic equations, it helps to analyze and use these properties to simplify or verify the expressions correctly.

Properties of Logarithms

Logarithms possess several properties that make them highly useful for manipulating and solving logarithmic equations. These properties transform complex problems into simpler forms, thus enabling easier computation and verification.

• Product Property: \( \ln(a \times b) = \ln a + \ln b \). This means if you multiply two numbers and take the natural logarithm of the result, it's equivalent to taking the natural logarithm of each number and adding those results.
• Quotient Property: \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \). With division, the logarithm of a quotient turns into the difference of the logarithms.
• Power Property: \( \ln(a^b) = b \ln a \). This property allows you to take the exponent and multiply it by the logarithm of the base, simplifying power expressions.

In our specific exercise, we see these properties in action. The original equation suggests \( \ln(x+1) = \ln x + \ln 1 \). By using the fact that \( \ln 1 = 0 \), the equation simplifies to \( \ln(x+1) = \ln x \), which is false unless \( x + 1 = x \). Utilizing these logarithmic properties appropriately ensures accurate solutions and allows adjustments to reach valid conclusions.

Equation Verification

Equation verification involves checking whether a mathematical statement or equation holds true by investigating and testing each component. When verifying equations with logarithms, one should apply known properties and rules. First, examine each part of the equation. For example, the equation \( \ln(x+1) = \ln x + \ln 1 \) needs scrutiny. Knowing that \( \ln 1 = 0 \), rewrite as \( \ln(x+1) = \ln x \). This equation suggests \( x + 1 = x \), which is incorrect. Therefore, the original statement is false. Correcting the equation requires consistent equalities on both sides based on logarithmic rules. By understanding that \( \ln(x+1) = \ln (x \cdot 1) \), applying the product property would confirm \( \ln(x) + \ln(1) \), ultimately rendering a truthful equation when proper rules are followed. Verification helps ensure accuracy and aids in corrections if mistakes are found, thus deepening your understanding of the concepts involved.