Question

If \(x^{3 / 4\left(\log _{3} x\right)^{2}+\log _{3} x-5 / 4}=\sqrt{3}\) then \(x\) can not assume (1) one positive integral value (2) one irrational value (3) two positive rational values (4) two distinct positive integral values

Short answer

(2) one irrational value

Step-by-step solution

Simplify the Equation

Start by simplifying the given equation. The original equation is \[ x^{\frac{3}{4} (\log_{3}{x})^2 + \log_3{x} - \frac{5}{4}} = \sqrt{3} \]Rewrite \( \sqrt{3} \) as \( 3^{1/2} \). So, the equation becomes\[ x^{\frac{3}{4} (\log_{3}{x})^2 + \log_3{x} - \frac{5}{4}} = 3^{1/2} \]

Equate Exponents

Since the bases are different, equate the exponents of the terms. Let \( y = \log_3{x} \). Then, recall that if\[ a^{f(x)} = a^{g(x)}, \text{ then } f(x) = g(x) \]Applying this concept, we get\[ \frac{3}{4} y^2 + y - \frac{5}{4} = \frac{1}{2} \]

Solve the Quadratic Equation

Rearrange the equation to form a standard quadratic equation:\[ \frac{3}{4} y^2 + y - \frac{5}{4} - \frac{1}{2} = 0 \]Simplify further by combining like terms:\[ \frac{3}{4} y^2 + y - \frac{3}{4} = 0 \]Multiply the entire equation by 4 to clear the fraction:\[ 3y^2 + 4y - 3 = 0 \]

Factor the Quadratic Equation

Factor the quadratic equation \[ 3y^2 + 4y - 3 = 0 \]Find factors that satisfy the equation by trial and checking or using the quadratic formula: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In this case: \[ y = \frac{-4 \pm \sqrt{16 + 36}}{6} = \frac{-4 \pm \sqrt{52}}{6} = \frac{-4 \pm 2\sqrt{13}}{6} = \frac{-2 \pm \sqrt{13}}{3} \]

Convert Back to Original Variable

Recall that \( y = \log_3{x} \). Therefore,\[ \log_3{x} = \frac{-2 + \sqrt{13}}{3} \text{ or }\log_3{x} = \frac{-2 - \sqrt{13}}{3} \]Since \( x \) must be positive,\[ x = 3^{\left( \frac{-2 + \sqrt{13}}{3} \right)} \]

Analyze the Values

Consider the values for \( x \). Here, \(x\) is derived from \( y = \log_3{x} \), and both values of \( y \) do not yield positive integral values. Hence, \( x \) can be one irrational value.Thus, the solution is:(1) one positive integral value - Wrong(2) one irrational value - Correct(3) two positive rational values - Wrong(4) two distinct positive integral values - Wrong

Key concepts

logarithmic_equations

Logarithmic equations are equations that involve logarithms of variables or constants. The equation usually takes the form:

\( a^{f(x)} = a^{g(x)} \)
In logarithmic equations, it is often helpful to convert the logarithmic form into an exponential form. This technique simplifies solving the equation. In our exercise, we applied this method by setting \( y = \log_3{x} \).

This not only made the equation easier to handle but also aided in finding the solutions step-by-step. Remember, the properties of logarithms can turn complex equations into more manageable forms. Ensure you're comfortable with these properties, such as the power rule, product rule, and change of base formula.

quadratic_equations

A quadratic equation is a second-order polynomial equation in a single variable \( x \) with the form:

\( ax^2 + bx + c = 0 \)
To solve quadratic equations, there are several methods, including factoring, using the quadratic formula, and completing the square. In our exercise, we simplified the equation into a quadratic form:

\( 3y^2 + 4y - 3 = 0 \)
Here, we used the quadratic formula:

\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This gave us the solutions for \( y \). Understanding how to solve quadratic equations is essential because they appear frequently in different branches of algebra and calculus.

irrational_numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction or a ratio of two integers. They have non-repeating, non-terminating decimal expansions. Examples include \( \sqrt{2} \), \( \pi \), and the solutions obtained in our example:

\( y = \frac{-2 + \sqrt{13}}{3} \).

When we analyzed the values of \( x \), we saw that this resulted in:

\( x = 3^{(\frac{-2 + \sqrt{13}}{3})} \)
This confirms that \( x \) is an irrational number, as it can't be simplified into a ratio of two integers. Understanding irrational numbers is vital because they often appear as solutions in various mathematical contexts.

exponents

Exponents are a shorthand way to express repeated multiplication of the same number. The notation \( a^b \) means \( a \) is multiplied by itself \( b \) times. In the exercise, we dealt extensively with exponents:

\( x^{\frac{3}{4}(\log_3 x)^2 + \log_3 x - \frac{5}{4}} = 3^{1/2} \)

This was simplified using properties of exponents, showing how crucial a strong understanding of these properties is. Key properties include:

• The product rule: \( a^m * a^n = a^{m+n} \)
• The power rule: \( (a^m)^n = a^{mn} \)
• The quotient rule: \( \frac{a^m}{a^n} = a^{m-n} \)

These properties aid the simplification and solution of complex algebraic problems. Mastering exponents will empower you to tackle a wide range of mathematical challenges.