Question

Solve each equation. $$ 2^{-x}=16 $$

Short answer

x = -4

Step-by-step solution

Understand the Problem

The given equation is an exponential equation with a base of 2. It is given as: \[2^{-x}=16\]

Rewrite 16 as a Power of 2

Recognize that 16 can be written as a power of 2. Since \[16 = 2^4\]we can substitute this into the equation: \[2^{-x} = 2^4\]

Set the Exponents Equal to Each Other

Since the bases are the same, we can set the exponents equal to each other: \[-x = 4\]

Solve for x

To isolate x, divide or multiply both sides by -1: \[x = -4\]

Key concepts

Solving Exponential Equations

When solving exponential equations, it's essential to recognize equations where the variable appears in the exponent. To solve these, you'll often need to employ a few techniques like expressing both sides of the equation with the same base. This can simplify the equation and make it easier to solve.
In our example, the equation is \(2^{-x}=16\). First, rewrite 16 as a power of 2, which is \(2^4\). Then, you get \(2^{-x}=2^4\). Since the bases are identical, set the exponents equal to each other: \(-x=4\). Solve for \(x\) by multiplying both sides by -1, giving you \(x=-4\).
This approach of isolating the variable by matching bases and setting exponents equal is a powerful tool for solving exponential equations efficiently.

Exponents

Exponents are shorthand for repeated multiplication of a number by itself. For instance, \(2^4\) means multiplying 2 by itself 4 times: \(2×2×2×2 = 16\). Exponents follow specific rules, such as:
- **Multiplication Rule:** \(a^m \times a^n = a^{m+n}\)
- **Division Rule:** \(a^m / a^n = a^{m-n}\)
- **Power Rule:** \((a^m)^n = a^{m×n}\)
Exponents can also be negative or fractional. A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent, while a fractional exponent represents roots. For example, \(2^{-3} = 1/2^3 = 1/8\) and \(16^{1/2} = 4\) (the square root of 16). Understanding these rules helps simplify and solve equations involving exponents.

Properties of Exponents

Properties of exponents are crucial for simplifying and solving equations involving exponents. These properties include:
- **Zero Exponent:** Any non-zero base raised to the power of zero equals 1. For instance, \(5^0 = 1\).
- **Negative Exponent:** A negative exponent signifies the reciprocal of the base raised to the absolute value of the exponent. For example, \(3^{-2} = 1/3^2 = 1/9\).
- **Product of Powers:** When multiplying two exponents with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\).
- **Quotient of Powers:** When dividing two exponents with the same base, subtract the exponents: \(a^m / a^n = a^{m-n}\).
- **Power of a Power:** When raising an exponent to another power, multiply the exponents: \((a^m)^n = a^{m×n}\).
Familiarity with these properties will allow you to manipulate and solve exponential equations more effectively.