If $5^{3 s-1}=5^{b-3}$, what is the value of $b$?
Answer (1)
Equate the exponents: 3 s − 1 = b − 3 . Solve for b : b = 3 s + 2 . Test the given options for b to see if they yield a valid s . If b = 1 , then s = − 1/3 , which is a valid solution. 1
Explanation
Equating the exponents We are given the equation 5 3 s − 1 = 5 b − 3 and asked to find the value of b . Since the bases are equal, we can equate the exponents.
Setting up the equation Equating the exponents, we have 3 s − 1 = b − 3 .
Solving for b Now, we solve for b in terms of s . Adding 3 to both sides of the equation, we get 3 s − 1 + 3 = b , which simplifies to b = 3 s + 2 .
Finding possible values for b Since the problem asks for the value of b , but does not provide a value for s , we cannot find a specific numerical value for b . However, we can express b in terms of s as b = 3 s + 2 .
Looking at the provided options, we can see if any of them can be obtained by plugging in a value for s .
If b = − 2 , then 3 s + 2 = − 2 , so 3 s = − 4 , and s = − 3 4 .
If b = 1 , then 3 s + 2 = 1 , so 3 s = − 1 , and s = − 3 1 .
If b = 2 , then 3 s + 2 = 2 , so 3 s = 0 , and s = 0 .
Since the problem does not specify a value for s , we cannot determine a unique value for b . However, if we assume that s is such that b = 1 , then 3 s − 1 = − 2 and b − 3 = 1 − 3 = − 2 , so 3 s − 1 = b − 3 holds. Therefore, b = 1 is a possible value for b .
Determining the value of b The problem states that 5 3 s − 1 = 5 b − 3 . Therefore, the exponents must be equal: 3 s − 1 = b − 3 . We want to find the value of b . Solving for b , we have b = 3 s − 1 + 3 = 3 s + 2 . We are given the possible values of b as − 2 , 1 , and 2 . If b = − 2 , then 3 s + 2 = − 2 , so 3 s = − 4 , which means s = − 4/3 . If b = 1 , then 3 s + 2 = 1 , so 3 s = − 1 , which means s = − 1/3 . If b = 2 , then 3 s + 2 = 2 , so 3 s = 0 , which means s = 0 . Since s can take on any value, we need to determine which of the given values of b is possible. If we choose b = 1 , then 3 s − 1 = 1 − 3 = − 2 , so 3 s = − 1 , and s = − 1/3 . Therefore, b = 1 is a possible value.
Examples
In exponential equations, when the bases are the same, we can equate the exponents to solve for unknown variables. This principle is used in various fields, such as calculating growth rates in finance or determining decay rates in radioactive materials. For example, if you have an investment that grows at a certain rate, and you want to know how long it will take to reach a specific value, you can set up an exponential equation and solve for the time variable using the equality of exponents.
