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Chapter 4: Problem 81

Let \(f(x)=2^{x}, g(x)=(1 / 3)^{x}, h(x)=10^{x},\) and \(m(x)=e^{x} .\) Find the value of \(x\) in each equation. $$f(x)=4$$

Short Answer

Expert verified
x = 2

Step by step solution

01

- Write down the equation

The given equation is f(x) = 4 where f(x) = 2^x.
02

- Set the function equal to the given value

Substitute f(x) with 2^x to get: 2^x = 4.
03

- Express 4 as a power of 2

Note that 4 = 2^2. So, replace 4 with 2^2 in the equation: 2^x = 2^2.
04

- Equate the exponents

Since the bases are the same, the exponents must be equal too: x = 2.

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Key Concepts

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

Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. For instance, in the function f(x) = 2^x, 2 is the base and x is the exponent. These functions are used widely in different fields due to their rapid growth or decay. They are useful in modeling scenarios involving population growth, radioactive decay, and more. Recognizing how to manipulate these expressions is crucial in solving exponential equations. Always remember, the fundamental parts of an exponential function involve the base (a constant) and the exponent (a variable).
Equating Exponents
When you encounter an equation with the same base on both sides, such as 2^x = 2^2, you can directly equate the exponents. This step simplifies solving the equation significantly. Recognizing that identical bases imply the exponents must be equal allows us to focus solely on the variable parts. Therefore, knowing that 2^x = 2^2 implies x = 2 simplifies the process.
Powers of a Number
Understanding powers of a number is vital. A power represents repeated multiplication of a number by itself. For example, 2^2 indicates 2 脳 2 = 4, and 2^3 indicates 2 脳 2 脳 2 = 8. Recognizing these common powers helps simplify and solve exponential equations. Memorize a few lower powers of base numbers like 2, 3, 5, and 10, since they frequently appear in algebraic expressions. In the given exercise, knowing that 4 can be expressed as 2^2 made equating the exponents straightforward.
Function Notation
Functions have specific notations that help identify their behaviors. For instance, f(x) = 2^x is read as 'f of x equals 2 raised to the power of x.' Function notation allows us to define complex relationships in a concise manner. It helps in specifying inputs and corresponding outputs clearly. While working with functions in equations, substituting given values correctly is essential. For example, when the problem states f(x) = 4, you replace f(x) with its definition 2^x, making it easier to proceed with solving the equation. Function notation essentially serves as shorthand for expressing dependencies between variables and their computations.

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Most popular questions from this chapter

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