Problem 14
Question
Determine the number that can be used in place of the question mark to make the equation true. $$16^{?}=16$$
Step-by-Step Solution
Verified Answer
The value of the exponent is 1.
1Step 1: Understand the Equation
The given equation is \(16^{?}=16\). This means we need to determine the value of the exponent that makes the base 16 equal to 16.
2Step 2: Rewrite the Base Term
Rewrite the number 16 in exponential form. We know that \(16 = 16^1\).
3Step 3: Equate the Exponents
Since both sides of the equation have the same base (16), we can set the exponents equal to each other: \( ? = 1 \).
Key Concepts
Base and exponentExponential equationsEquating exponents
Base and exponent
To understand exponential equations, we first need to understand the terms 'base' and 'exponent'. An exponential expression takes the form of \(a^b\), where \(a\) is the base and \(b\) is the exponent.
The base in an exponential expression is the number that gets multiplied. The exponent tells us how many times the base is multiplied by itself. For example, in \(2^3\), 2 is the base, and 3 is the exponent.
The base in an exponential expression is the number that gets multiplied. The exponent tells us how many times the base is multiplied by itself. For example, in \(2^3\), 2 is the base, and 3 is the exponent.
- \(2^3 = 2 \times 2 \times 2 = 8\)
Exponential equations
An exponential equation is an equation in which the variables appear as exponents. An example is \(16^?=16\).
To solve such equations, we need to manipulate the equation until both sides have the same base. Once the bases are the same, we can equate the exponents.
Here is a simple example:
Suppose we have \(2^x = 8\). We know that 8 can be written as \(2^3\). Therefore, \(2^x = 2^3\). Since the base (2) is the same, we can equate the exponents:
To solve such equations, we need to manipulate the equation until both sides have the same base. Once the bases are the same, we can equate the exponents.
Here is a simple example:
Suppose we have \(2^x = 8\). We know that 8 can be written as \(2^3\). Therefore, \(2^x = 2^3\). Since the base (2) is the same, we can equate the exponents:
- \(x = 3\)
Equating exponents
Equating exponents is the final step in solving exponential equations.
Once we have the same base on both sides of the equation, we can simply set the exponents equal to each other.
For the original problem, \(16^?=16\), we first recognize that 16 can be rewritten in exponential form as \(16^1\).
Therefore, our equation becomes \(16^? = 16^1\). Since the bases are the same (16), we can equate the exponents on both sides:
This straightforward approach makes solving such problems more manageable once you grasp the importance of equating exponents.
Once we have the same base on both sides of the equation, we can simply set the exponents equal to each other.
For the original problem, \(16^?=16\), we first recognize that 16 can be rewritten in exponential form as \(16^1\).
Therefore, our equation becomes \(16^? = 16^1\). Since the bases are the same (16), we can equate the exponents on both sides:
- \(? = 1\)
This straightforward approach makes solving such problems more manageable once you grasp the importance of equating exponents.
Other exercises in this chapter
Problem 13
Evaluate each exponential expression without using a calculator. $$2^{-3}$$
View solution Problem 14
Rewrite each expression as a single logarithm. $$\log _{3}(x+2)+\log _{3}(x-1)$$
View solution Problem 14
Evaluate each exponential expression without using a calculator. $$3^{-2}$$
View solution Problem 15
Each of these equations involves more than one logarithm. Solve each equation. Give exact solutions. $$\log _{2}(x+2)+\log _{2}(x-2)=5$$
View solution