Problem 56 - Algebra and Trigonometry

Question

For what value of \(x\) is it true that \((\log x)^{3}=3 \log x ?\)

Step-by-Step Solution

Answer

The values of \(x\) are 1, \(10^{\sqrt{3}}\), and \(10^{-\sqrt{3}}\).

Step 1: Set the Equation

Start with the equation given: \((\log x)^{3}=3 \log x\). We need to find the value of \(x\) that satisfies this equation.

Step 2: Simplify by Substitution

Let \( y = \log x \). This simplifies our equation to \(y^3 = 3y\).

Step 3: Factor the Polynomial

The equation \(y^3 = 3y\) can be rewritten as \(y^3 - 3y = 0\). Factoring out \(y\), we get \(y(y^2 - 3) = 0\).

Step 4: Solve for y

Set each factor equal to zero to find possible values of \(y\).1. \(y = 0\)2. \(y^2 - 3 = 0\) which implies \(y^2 = 3\). Hence, \(y = \sqrt{3}\) or \(y = -\sqrt{3}\).

Step 5: Substitute Back for x

Recall \(y = \log x\), substitute back to find \(x\).1. If \(y = 0\), then \(\log x = 0\) implies \(x = 10^0 = 1\).2. If \(y = \sqrt{3}\), then \(\log x = \sqrt{3}\) implies \(x = 10^{\sqrt{3}}\).3. If \(y = -\sqrt{3}\), then \(\log x = -\sqrt{3}\) implies \(x = 10^{-\sqrt{3}}\).

Step 6: Verify Solutions

Check the original equation with each value of \(x\).1. \(x=1\): \((\log 1)^3 = 3\log 1\), both sides are 0, so it's true.2. \(x = 10^{\sqrt{3}}\): \((\sqrt{3})^3 = 3\sqrt{3}\), both simplify to \(3\sqrt{3}\), so it's true.3. \(x = 10^{-\sqrt{3}}\): \((-\sqrt{3})^3 = -3\sqrt{3}\) and the right side is also \(-3\sqrt{3}\), so it's true.

Key Concepts

Polynomial Factoring

When dealing with equations involving polynomials, factoring plays a crucial role. Polynomial factoring is the process of expressing a polynomial as a product of its factors. This helps in solving polynomial equations by breaking them down into simpler ones.

Let's consider the equation given: \[ y^3 - 3y = 0 \] This equation is a polynomial in terms of \(y\). To factor it, we find common terms. In this case, \(y\) is a common factor. Factor \(y\) out of each term:

• \(y \times (y^2 - 3) = 0\)

Now, we have factored the polynomial into two parts: \(y\) and \((y^2 - 3)\).

This step transforms a cubic equation into simpler linear and quadratic forms. By setting each factor equal to zero, we find the possible values of \(y\). This technique helps simplify the problem significantly.

Logarithm Properties

Understanding logarithms is essential for solving the original exercise. Logarithms are the inverse operations of exponentiation. They have valuable properties that simplify complex equations.

One key property of logarithms is the change of base formula, which is implicitly used when solving equations like \((\log x)^3 = 3 \log x\). Additionally, knowing that \(\log_{10} 10 = 1\) can be helpful when calculating solutions for logarithmic equations.
Logarithm properties also provide tools like:

• \(\log_a (MN) = \log_a M + \log_a N\)
• \(\log_a \left(\frac{M}{N}\right) = \log_a M - \log_a N\)
• \(\log_a M^k = k\log_a M\)

In our exercise, recognizing that \(\log x = y\) simplifies the equation considerably, turning logarithmic challenges into more familiar polynomial ones.

Substitution Method

The substitution method is a clever technique that simplifies complex equations by temporarily introducing a new variable. It is particularly useful in solving equations where a direct approach might be cumbersome or unclear.

In the original exercise, substitution plays a pivotal role. By letting \(y = \log x\), the original logarithmic equation \((\log x)^3 = 3\log x\) transforms into a polynomial equation \(y^3 = 3y\).
This reduces the problem to a more conventional form, easier to manipulate and solve. Breaking down the complex problem into manageable pieces is one of the strengths of substitution. Once you solve for \(y\), simply reverse the substitution to find \(x\).

This method streamlines the problem-solving process and allows us to understand and solve equations more efficiently. It is a versatile tool, especially beneficial when dealing with composite functions or nested operations.