/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 53 Solve for \(x: 2^{2 / \log _{3} ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 53

Solve for \(x: 2^{2 / \log _{3} x}=\frac{1}{16}\)

Short Answer

Expert verified
\(x = \frac{1}{\sqrt{3}}\)

Step by step solution

01

Rewrite the Equation

Start by rewriting the given equation \(2^{2 / \log_{3} x} = \frac{1}{16}\). The right side, \(\frac{1}{16}\), can be expressed as a power of 2. Notice that \(\frac{1}{16} = 2^{-4}\). Thus, the equation becomes \(2^{2 / \log_{3} x} = 2^{-4}\).
02

Equate the Exponents

Since the bases on both sides of the equation are the same, you can equate the exponents: \(\frac{2}{\log_{3} x} = -4\).
03

Solve for \( \log_{3} x \)

Multiply both sides by \(\log_{3} x\) to get: \(2 = -4 \cdot \log_{3} x\). Then, divide both sides by \(-4\) to isolate \(\log_{3} x\): \(\log_{3} x = -\frac{1}{2}\).
04

Solve for \(x\) Using Properties of Logarithms

Recall that \(\log_{3} x = y\) implies \(x = 3^{y}\). So if \(\log_{3} x = -\frac{1}{2}\), then \(x = 3^{-\frac{1}{2}}\).
05

Simplify the Expression for \(x\)

Express \(x = 3^{-\frac{1}{2}}\) as \(x = \frac{1}{\sqrt{3}}\). This is the simplified form of the solution for \(x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Equations
Exponential equations are equations where variables appear as exponents. They often have the form \(a^{f(x)} = a^{g(x)}\). To solve exponential equations:
  • First, express both sides with the same base if possible. This often makes it easier to equate the exponents.
  • If both sides of the equation have the same base, set the exponents equal to each other.
For example, in the equation \(2^{2 / \log_{3} x} = \frac{1}{16}\), we first converted \(\frac{1}{16}\) to \(2^{-4}\) because they share the base 2. Once the bases match, the exponents can be set equal, facilitating a transition to solving the equation.
Logarithms
Logarithms are the inverse operations of exponentials. They help in solving equations where the unknown value is an exponent. The logarithm \(\log_b y\) answers the question: "To what power must we raise \(b\) to get \(y\)?"
  • If \(b^y = x\), then \(\log_b x = y\).
  • Logarithms transform multiplicative relationships into additive ones, aiding in simplifying complex calculations.
In solving \(2^{2 / \log_{3} x} = \frac{1}{16}\), we needed to understand that \(\log_{3} x = \log_3 (3^{- rac{1}{2}})\) implies \(x = 3^{\log_3 x}\). This understanding of logarithms helps transition from the logarithmic form to solving for \(x\) directly.
Properties of Exponents
The properties of exponents are crucial for manipulating and simplifying exponential expressions. Some key properties include:
  • \(a^m \, \cdot \, a^n = a^{m+n}\)
  • \((a^m)^n = a^{m \cdot n}\)
  • \(a^{-n} = \frac{1}{a^n}\)
In the given problem, converting \(\frac{1}{16}\) to \(2^{-4}\) utilized the property of negative exponents, which states that \(a^{-n} = \frac{1}{a^n}\). Moreover, rewriting \(x = 3^{- rac{1}{2}}\) as \(x = \frac{1}{\sqrt{3}}\) uses the understanding that exponents can represent roots, further simplifying the expression for practical applications.

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