Problem 94 - College Algebra

Question

In Exercises \(89-102,\) 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. After evaluating both sides of the equation, it is found that \(\ln (x+1) \neq \ln x\). In order to make the equation true, it should be modified to \(\ln(x+1) = \ln(x+1)\).

Step 1: Analyze and simplify the equation

Evaluate the right-hand side of the equation \(\ln x + \ln 1\). Recall that any number to the power of 0 is 1, therefore \(\ln 1 = 0\). Substitute this into the equation to get \(\ln x + 0\), which simplifies to \(\ln x\).

Step 2: Evaluate the left-hand side of the equation

Evaluate the left-hand side of the equation, which is \(\ln (x + 1)\). There is no mathematical property that can simplify this expression. Therefore, it remains as is \(\ln (x + 1)\).

Step 3: Comparison of both sides

Compare both sides of the equations. According to the performances made, the simplified equation is found as \(\ln (x+1) = \ln x\). Since the right and left sides are not the same, it can be stated that the initial equation is incorrect.

Step 4: Modify the initial equation

To make the initial equation true, replace \(\ln 1\) with \(\ln (x + 1) - \ln x\). So, the correct equation should be \(\ln (x+1) = \ln x + \ln (x + 1) - \ln x\), which simplifies to \(\ln(x+1) = \ln(x+1)\).

Key Concepts

Natural Logarithm Properties

When dealing with natural logarithm properties, we're focusing on the logarithm to the base 'e', where 'e' is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm of a number 'x' is commonly written as \( \ln(x) \).

Here are some important properties:

• The natural logarithm of 1, \( \ln(1) \) is 0 because 'e' to the power of 0 equals 1.
• For any positive numbers 'a' and 'b', \( \ln(a \cdot b) \) is equal to \( \ln(a) + \ln(b) \) due to the law of logarithms about product.
• The natural logarithm of 'e' is 1 because 'e' to the power of 1 equals 'e' itself.

This property is indispensable when simplifying and transforming logarithmic expressions in algebra. Understanding these properties ensures that we can evaluate and manipulate logarithmic equations effectively.

Evaluating Logarithms

Evaluating logarithms is the process of finding the power to which the base of the logarithm must be raised to obtain a given number. For natural logarithms, the base is 'e'.

To evaluate a logarithm like \( \ln(x) \), one might sometimes find equivalent expressions or use properties of logarithms. For instance, \( \ln(x^y) \) is \( y \cdot \ln(x) \) and \( \ln(e) = 1 \). Evaluating logarithms can involve simplification, conversion between logarithmic and exponential forms, and using the properties and identities of logarithms to simplify complex expressions. This process is essential when solving logarithmic equations in algebra.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations—addition, subtraction, multiplication, division—and, in more advanced cases, other operations such as logarithms and exponents.

For example, \( \ln(x+1) - \ln(x) \) is an algebraic expression involving logarithms. These expressions follow the same rules and properties as simpler algebraic expressions, such as the distributive property, associative property, and others. Being able to manipulate these expressions correctly is pivotal for solving algebraic equations, including those involving logarithms.

True or False Equations

True or false equations are statements that claim equality between two expressions and can either be correct (true) or incorrect (false). In mathematics, it's crucial to be able to verify the truth of these equations. This often requires simplifying both sides of the equation and using algebraic properties.

If an equation turns out to be false, it's important to either identify and correct the mistake or disprove the equivalence entirely. In the example \( \ln (x+1)=\ln x+\ln 1 \), it is deemed false because when evaluated, it simplifies to \( \ln (x+1) eq \ln x \). Corrections are applied to make a true statement by understanding and applying logarithmic properties correctly.