Problem 21 - Contemporary Precalculus

Question

Use graphical or algebraic means to determine whether the statement is true or false. $$\ln x^{3}=(\ln x)^{3} ?$$

Step-by-Step Solution

Answer

Answer: False

Step 1: Simplify the left side of the equation

We can use the logarithmic property of exponents which states that $$\ln a^b = b\ln a.$$ Therefore, we can rewrite $$\ln x^3$$ as follows: $$\ln x^3 = 3\ln x.$$

Step 2: Check if the left side of the equation equals the right side

Now that we have simplified the left side of the equation, let's compare it to the right side: $$(\ln x)^3.$$ We obtained that the left side of the equation is $$3\ln x.$$ If the given statement were true, then $$3\ln x = (\ln x)^3.$$ However, these two expressions are not the same, as one is the cube of the logarithm and the other is three times the logarithm. Therefore, the given statement is

Step 3: false

.

Key Concepts

Natural Logarithm

The natural logarithm, denoted by \(\ln\), is a mathematical function that is particularly important in the world of calculus and precalculus. Thought of as the inverse operation of exponentiation when using Euler's number \((e)\), the natural log answers the question: 'To what power must \((e)\) be raised to obtain a given number?' For example, if we have \(\ln(x)\), we are seeking the power to which \((e)\) must be raised to get \(x\).

Natural logarithms have unique properties that make algebraic manipulations involving exponential and logarithmic equations easier. One such property, relevant to this exercise, is the logarithm of a power, which states that \(\ln(a^b) = b\ln(a)\). This simplifies the process of working with powers inside a logarithm and is pivotal to understanding the solution to our original problem. Additionally, the natural logarithm function is continuous and differentiable, making it a staple tool for solving calculus problems involving growth rates and areas under curves.

Exponent Rules

Exponent rules, also known as the laws of exponents, are the backbone of algebraic operations involving powers. These rules dictate how to manipulate expressions with exponents and are essential for simplifying and solving equations.

Some of the fundamental exponent rules include:

• The product rule: \(a^m \times a^n = a^{m+n}\)
• The quotient rule: \(\frac{a^m}{a^n} = a^{m-n}\)
• The power of a power rule: \(\left(a^m\right)^n = a^{mn}\)
• The zero exponent rule: \(a^0 = 1\), given \(a \eq 0\)
• The negative exponent rule: \(a^{-n} = \frac{1}{a^n}\)

The problem we are inspecting relates to a mixed application of logarithmic and exponent rules. It showcases the necessity to distinguish between the rules for pure exponents and those applicable when logarithms are also involved.

Algebraic Manipulation

Algebraic manipulation is the skill set that allows us to rearrange and simplify mathematical expressions effectively. It involves various operations such as factoring, expanding, and the use of identities to work with equations—all fundamental in precalculus and higher-level mathematics.

Key techniques of algebraic manipulation include:

• Distributive property: \(a(b + c) = ab + ac\)
• Commutative property: \(ab = ba\) and \(a + b = b + a\)
• Associative property: \(a(bc) = (ab)c\) and \(a + (b + c) = (a + b) + c\)
• Combining like terms: Grouping and simplifying terms that have the same variables raised to the same power.
• Factoring: Breaking down composite numbers or expressions into a product of their factors.

By mastering algebraic manipulation, students can simplify complex problems, such as the problem in the exercise, by recognizing when and how to apply the properties of logarithms and exponents effectively.