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Chapter 1: Problem 13

Solve for \(x\). $$ \left(\frac{1}{16}\right)^{x}=\frac{1}{8} $$

Short Answer

Expert verified
The solution is \(x = \frac{3}{4}\).

Step by step solution

01

Rewrite Bases as Powers of 2

The first step is to express both sides of the equation using a base of 2. We know:- \(\frac{1}{16} = 2^{-4}\) because \(16 = 2^4\).- \(\frac{1}{8} = 2^{-3}\) because \(8 = 2^3\).So we can rewrite the equation as:\((2^{-4})^x = 2^{-3}\)
02

Use the Laws of Exponents

Apply the power of a power property \((a^m)^n = a^{mn}\) to the left side:\(2^{-4x} = 2^{-3}\)
03

Equate the Exponents

Since the bases are the same (both base 2), set the exponents equal to each other:\(-4x = -3\)
04

Solve for x

Divide both sides of the equation by -4 to solve for \(x\):\( x = \frac{-3}{-4} = \frac{3}{4}\)

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Key Concepts

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

Laws of Exponents
The laws of exponents are fundamental rules that guide how we handle expressions involving powers. They make it easier to simplify and solve equations with exponents. There are several laws of exponents, but here are some key ones that often come into play:
  • Product of Powers: For any base \(a\), \(a^m \cdot a^n = a^{m+n}\). This means when we multiply like bases, we add their exponents.

  • Power of a Power: For any base \(a\), \((a^m)^n = a^{mn}\). When raising a power to another power, multiply the exponents.

  • Power of a Product: For a product of bases, \((ab)^m = a^m \cdot b^m\). This shows that each factor gets raised to the exponent separately.
In the original exercise, we used the power of a power rule: \((2^{-4})^x\) becomes \(2^{-4x}\). This rule is crucial when dealing with bases raised to multiple exponents.
Exponential Functions
Exponential functions are equations in which a constant base is raised to a variable exponent. They rapidly increase or decrease, and are often represented as \(y = a^x\), where \(a\) is a positive constant.These functions are important in various fields such as biology for modeling population growth and in finance for calculating compound interest. One of the main characteristics is the rapid rate of change.
  • Growth: If the base \(a > 1\), the function represents exponential growth.

  • Decay: If the base \(0 < a < 1\), the function represents exponential decay.
In solving the given exercise, understanding how to manipulate exponential functions is key to rewiring the equation for easier solving.
Rewriting Bases
Rewriting bases is a common technique used in solving exponential equations. It involves expressing both sides of an equation with the same base to simplify solving for the variable.In the exercise, the bases were rewritten from fractions to powers of 2. For instance, \(\frac{1}{16}\) was rewritten as \(2^{-4}\) since \(16 = 2^4\) and we take the reciprocal for negative exponents. Similarly, \(\frac{1}{8}\) was expressed as \(2^{-3}\).This technique allows us to equate the exponents directly since both sides become powers of the same base. Thus, rewriting bases is a powerful tool in solving exponential equations where the base isn't immediately obvious.

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