Problem 38 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

In Problems 38 through 44 find all \(x\) for which each equation is true. $$ [\log x]^{3}=\log \left(x^{3}\right) $$

Step-by-Step Solution

Answer

The possible solutions for \(x\) are \(10^{\sqrt[3]{3}}\) and \(10^{-\sqrt[3]{3}}\).

Step 1: Apply Logarithm Rule

Using the rule \(\log a^n = n \log a\), simplify the right-hand side to obtain: \([log x]^3 = 3 \log x\).

Step 2: Solve for \(x\)

First, square root both sides to get rid of the cube, resulting in \(| \log x | = \sqrt[3]{3 \log x}\). Divide both sides by \(\log x\) (we can do this because \(x > 0\), so \(\log x \neq 0\)) to obtain \(1 = \sqrt[3]{3}\). This means that \(\log x = \sqrt[3]{3}\) or \(-\log x = \sqrt[3]{3}\). Therefore, \(x = 10^{\sqrt[3]{3}}\) or \(x = 10^{-\sqrt[3]{3}}\).

Step 3: Determine the Valid Solution(s)

According to the logarithm's domain, \(x\) can only be a positive number. Checking both solutions, we can see that \(10^{\sqrt[3]{3}}\) is positive, but \(10^{-\sqrt[3]{3}}\) is positive too (as it's a positive fraction). So, both solutions are correct.

Key Concepts

Understanding Logarithms

Logarithms are an integral part of mathematics, used primarily to solve exponential equations. A logarithm answers the question: "To what exponent must a base be raised to produce a certain number?" When we see an equation like \(log_b a = c\), it implies that \(b^c = a\). This concept is central to converting between different forms of equations.
Logarithmic functions have some rules and properties that simplify their use:

• Product Rule: \(log_b (mn) = log_b m + log_b n\)
• Quotient Rule: \(log_b \left( \frac{m}{n} \right) = log_b m - log_b n\)
• Power Rule: \(log_b (m^n) = n log_b m\)

Understanding these properties allows us to manipulate and simplify complex logarithmic expressions, which is key when solving equations like the ones in your exercise.

Exponents and Their Role

Exponents are numbers or variables, denoting the "power" to which a base, typically a real number, is raised. In mathematical notation, \(b^n\) implies that the base \(b\) is multiplied by itself \(n\) times.
Exponentiation has several important properties:

• Multiplication of Exponents: \(b^m \cdot b^n = b^{m+n}\)
• Division of Exponents: \(\frac{b^m}{b^n} = b^{m-n}\)
• Power of a Power: \((b^m)^n = b^{m cdot n}\)

In logarithmic equations, understanding exponents is crucial as logarithms are essentially the inverse operation of exponentiation. This duality helps us solve equations by switching between exponential and logarithmic forms when needed.

Illustrating Solution Verification

Verification is a pivotal step in solving mathematical equations as it ensures that the solution obtained is accurate and valid. In the case of logarithmic equations, once a potential solution is found, verifying involves plugging the solution back into the original equation to check for consistency.
For your exercise, after finding that \(x = 10^{\sqrt[3]{3}}\) and \(x = 10^{-\sqrt[3]{3}}\) are solutions, you should substitute these values back into the equation. This step checks whether both sides of the equation remain equal, thereby verifying the correctness.

• If substituting the solution into the equation holds true, the solution is valid.
• If not, reconsider the mathematical steps involved or potential constraints on \(x\).

Verification confirming a solution respects all properties and constraints, like the non-negativity required in logarithmic functions.

Real Numbers in Equation Solutions

Real numbers encompass a broad set of numbers, including whole numbers, fractions, and irrational numbers. They are crucial because they allow for precise solutions in equations across a range of mathematical problems.
When solving logarithmic equations, it's important to determine if your solutions are real numbers and meet the necessary conditions of the logarithmic domain:

• Logarithms of non-positive numbers (like zero or negative numbers) are undefined, so solutions must be positive real numbers.
• This necessity ensures that only numbers from the set of positive real numbers can serve as valid solutions.

The concept of real numbers further enriches our understanding and verification in solving and interpreting the solution set to mathematical equations, therefore ensuring clarity and precision in mathematical problem-solving.