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Chapter 13: Problem 12

Evaluate the exponential function \(y=4^{x}\) for the given values of \(x\). $$x=4$$

Short Answer

Expert verified
When \( x = 4 \), \( y = 256 \).

Step by step solution

01

Understand the Expression

The expression given is an exponential function: \( y = 4^x \). This function describes how the variable \( y \) changes when \( x \) varies. We are tasked with finding \( y \) for the specific value \( x = 4 \).
02

Substitute the Given Value of x

Substitute \( x = 4 \) into the equation \( y = 4^x \):\[y = 4^4\]
03

Calculate the Exponent

Perform the calculation for \( 4^4 \). This means multiplying 4 by itself 4 times: \[4^4 = 4 \times 4 \times 4 \times 4\]
04

Carry Out the Multiplication

First, calculate \( 4 \times 4 = 16 \). Next, multiply the result by 4: \( 16 \times 4 = 64 \). Finally, multiply this product by 4 again to get:\[64 \times 4 = 256\]
05

Write the Final Result

Thus, the value of the exponential function \( y = 4^x \) when \( x = 4 \) is \( y = 256 \).

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Key Concepts

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

Substitution Method
The substitution method is a powerful tool used in mathematics to simplify expressions by replacing a variable with a given value. In the context of evaluating exponential functions like \( y = 4^x \), substitution involves plugging in a specific value for \( x \) into the function. This method helps simplify the process of solving equations by reducing the expression to more manageable numbers.

To use the substitution method, follow these steps:
  • Identify the variable being replaced (in this case, \( x \)).
  • Substitute the variable with the given numerical value. For this exercise, we replace \( x \) with 4.
  • Rewrite the expression with the new value in place of the variable, resulting in \( y = 4^4 \).
Substituting allows us to progress to calculation without ambiguity. It clarifies exactly what needs to be computed, ensuring accuracy in further numerical evaluations.
Exponentiation Process
The exponentiation process is central to understanding exponential functions. It involves raising a base number, called the "base" (here, it's 4), to a certain power (an exponent), which indicates how many times the base is used as a factor. In our example, the base is 4, and the exponent here is 4, shown as \( 4^4 \).

Exponentiation can be viewed as repeated multiplication:
  • The expression \( 4^4 \) means multiplying 4 by itself a total of four times: \( 4 \times 4 \times 4 \times 4 \).
  • This process makes it quicker to express and calculate very large numbers using a concise format.
Understanding the exponentiation process is essential for tackling problems involving powers, as it forms the basis for both simple and complex calculations. It鈥檚 a fundamental skill in mathematics, making computations on powers more efficient and manageable.
Calculation Steps
Performing calculations for exponential functions is straightforward once you follow a structured approach. In our example, we aim to find \( y \) by evaluating \( 4^4 \). Let's break it down into calculation steps for clarity:

  • First, compute \( 4 \times 4 \). This gives 16.
  • Next, take this result and multiply it by 4 again: \( 16 \times 4 = 64 \).
  • Finally, multiply the result by 4 one last time: \( 64 \times 4 = 256 \).
These steps show how exponential values grow rapidly, a characteristic property of exponential functions. Through systematic multiplication, we find that the initial expression \( y = 4^4 \) results in \( y = 256 \). Understanding and applying each calculation step results in clear and accurate outcomes, essential for mastery over topics involving exponential growth.

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Most popular questions from this chapter

\(\text {Solve the given problems.}\) A container of water is heated to \(90^{\circ} \mathrm{C}\) and then placed in a room at \(=0^{\circ} \mathrm{C} .\) The temperature \(T\) of the water is related to the time \(t\) (in \(\min\) ) Thy \(\log _{e} T=\log _{e} 90.0-0.23 t .\) Find \(T\) as a function of \(t\).

Find the natural antilogarithms of the given logarithms. $$-8.04 \times 10^{-3}$$

Find \(f(\text { in } \mathrm{Hz})\) if \(\ln f=21.619,\) where \(f\) is the frequency of the microwaves in a microwave oven.

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Perform the indicated operations. For a radioactive isotope, the constant of proportionality \(k\) is given by \(k=\frac{\log _{e}\left(^{1 / 2}\right)}{\text { half-life }} .\) Cobalt- \(60,\) used in cancer therapy, has a half-life of 5.27 years. What is the value of \(k ?\)
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