Problem 10
Question
Determine the number that can be used in place of the question mark to make the equation true. $$2^{?}=16$$
Step-by-Step Solution
Verified Answer
? = 4
1Step 1: Understand the equation
The equation given is \(2^{?} = 16\). The task is to find the exponent that makes this equation true.
2Step 2: Express 16 as a power of 2
Recognize that 16 can be expressed as a power of 2. We need to find which power of 2 results in 16. In other words, identify \(n\) such that \(2^n = 16\).
3Step 3: Identify the power of 2
Know that \(16 = 2^4\). Therefore, the exponent \(4\) makes the equation \(2^{?} = 16\) true.
4Step 4: Verify the solution
Double-check the work by substituting the exponent back into the equation: \(2^4 = 16\). Since the left side equals the right side, the solution is confirmed.
Key Concepts
exponentspowers of 2solving equations
exponents
An exponent refers to the number of times a number, known as the base, is multiplied by itself. In the equation given, we are dealing with exponential notation where the base is 2. For example, in the term 2^4, 2 is the base and 4 is the exponent. It tells us that 2 should be multiplied by itself 4 times: 2 × 2 × 2 × 2. This equals 16.
Understanding exponents is crucial in simplifying and solving various types of equations, including polynomial and logarithmic equations. Basic rules of exponents include:
Understanding exponents is crucial in simplifying and solving various types of equations, including polynomial and logarithmic equations. Basic rules of exponents include:
- Multiplying like bases: ewline \(a^m \times a^n = a^{m+n}\)
- Dividing like bases: ewline \(a^m \big/ a^n = a^{m-n}\)
- Power of a power: ewline \((a^m)^n = a^{m \times n}\)
powers of 2
Powers of 2 are specific examples of exponents where the base is always 2. These are very common in computer science and digital electronics. In our problem, we need to express 16 as a power of 2.
Common powers of 2 include:
Common powers of 2 include:
- \(2^0 = 1\)
- \(2^1 = 2\)
- \(2^2 = 4\)
- \(2^3 = 8\)
- \(2^4 = 16\)
- \(2^5 = 32\)
solving equations
Solving an equation means finding the value of the unknown variable that makes the equation true. Let's break down the steps used in solving our given exponential equation:
1. **Understand the equation:** What we need to find is the exponent that makes \(2^? = 16\) true.
2. **Express 16 as a power of 2:** Recognize that 16 can be written as \(2^4\). So, we are looking for the exponent \(n\) such that \(2^n = 16\).
3. **Identify the power:** We find that \(16 = 2^4\), which means n must be 4. Therefore, the exponent you're looking for is 4.
4. **Verify the solution:** Substitute 4 back into the equation to double-check. Since \(2^4 = 16\), our solution is correct.
By following these logical and straightforward steps, you can solve various exponential equations. Always verify your answers to ensure correctness.
1. **Understand the equation:** What we need to find is the exponent that makes \(2^? = 16\) true.
2. **Express 16 as a power of 2:** Recognize that 16 can be written as \(2^4\). So, we are looking for the exponent \(n\) such that \(2^n = 16\).
3. **Identify the power:** We find that \(16 = 2^4\), which means n must be 4. Therefore, the exponent you're looking for is 4.
4. **Verify the solution:** Substitute 4 back into the equation to double-check. Since \(2^4 = 16\), our solution is correct.
By following these logical and straightforward steps, you can solve various exponential equations. Always verify your answers to ensure correctness.
Other exercises in this chapter
Problem 9
Evaluate each exponential expression without using a calculator. $$3^{3}$$
View solution Problem 10
Simplify each expression. $$\log _{4}\left(2^{300}\right)$$
View solution Problem 10
Evaluate each exponential expression without using a calculator. $$2^{5}$$
View solution Problem 11
Rewrite each expression as a single logarithm. $$\log (5)+\log (3)$$
View solution