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Which of the following is equivalent to [tex]\( 4^3 = 64 \)[/tex]?

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]


Sagot :

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]