Identify the values of $a, b$, and $c$ that make the statement below true. [tex]
$\log _2 64=6$ if and only if $a^b=c$.
[/tex] [tex]
a=\square
b=\square
c=\square
[/tex]
Answer (2)
Rewrite the logarithmic equation in exponential form: lo g 2 64 = 6 becomes 2 6 = 64 .
Compare the exponential form with a b = c to identify the values.
Determine that a = 2 , b = 6 , and c = 64 .
State the final answer: a = 2 , b = 6 , c = 64 .
Explanation
Understanding the Problem We are given the logarithmic equation lo g 2 64 = 6 and we want to find the values of a , b , c such that this is equivalent to a b = c . In other words, we want to rewrite the logarithm in exponential form.
Converting to Exponential Form The logarithmic equation lo g 2 64 = 6 can be rewritten in exponential form as 2 6 = 64 . Comparing this with a b = c , we can identify the values of a , b , and c .
Identifying the Values By comparing 2 6 = 64 with a b = c , we can see that a = 2 , b = 6 , and c = 64 . Therefore, the values that make the statement true are a = 2 , b = 6 , and c = 64 .
Final Answer Thus, we have a = 2 , b = 6 , and c = 64 .
Examples
Logarithms and exponentials are used in many real-world applications, such as calculating the magnitude of earthquakes using the Richter scale, modeling population growth, and determining the half-life of radioactive materials. For example, if we know the initial amount of a radioactive substance and its half-life, we can use exponential decay to predict how much of the substance will remain after a certain amount of time. This is crucial in fields like nuclear medicine and environmental science.
The values that make the statement lo g 2 64 = 6 true when expressed as a b = c are a = 2 , b = 6 , and c = 64 .
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