To determine which of the given logarithmic forms is equivalent to the exponential equation [tex]\(4^3 = 64\)[/tex], let's recall the relationship between exponential and logarithmic forms.
The exponential form [tex]\(a^b = c\)[/tex] can be converted to the logarithmic form as [tex]\(\log_a(c) = b\)[/tex].
Given the equation [tex]\(4^3 = 64\)[/tex]:
- Here, the base [tex]\(a\)[/tex] is [tex]\(4\)[/tex].
- The exponent [tex]\(b\)[/tex] is [tex]\(3\)[/tex].
- The result [tex]\(c\)[/tex] is [tex]\(64\)[/tex].
Using the relationship, we convert [tex]\(4^3 = 64\)[/tex] to logarithmic form:
[tex]\[
\log_4(64) = 3
\][/tex]
Now, let’s examine the given options:
a. [tex]\(\log_4(64) = 3\)[/tex]
b. [tex]\(\log_3(4) = c\)[/tex]
c. [tex]\(\log_{64}(3) = 4\)[/tex]
d. [tex]\(\log_3(4) = 64\)[/tex]
From our conversion, we see that the correct logarithmic form is [tex]\(\log_4(64) = 3\)[/tex], which matches option a.
Therefore, the correct answer is:
a. [tex]\(\log_4(64) = 3\)[/tex]
