Answer :
To determine which logarithmic equation is equivalent to the expression [tex]\(2^7 = 128\)[/tex], let's first understand the relationship between exponential and logarithmic forms. When you have an exponential equation of the form [tex]\(a^b = c\)[/tex], it can be rewritten in its logarithmic form as [tex]\(\log_a(c) = b\)[/tex].
Let's apply this to [tex]\(2^7 = 128\)[/tex]:
1. Base (a): The base here is 2.
2. Exponent (b): The exponent is 7.
3. Result (c): The result is 128.
Using the logarithmic form [tex]\(\log_a(c) = b\)[/tex], substitute the values we know:
[tex]\[
\log_2(128) = 7
\][/tex]
Now compare this with the given options:
- a) [tex]\(\log_7 2 = 128\)[/tex]
- b) [tex]\(\log_2 7 = 128\)[/tex]
- c) [tex]\(\log_7 128 = 2\)[/tex]
- d) [tex]\(\log_2 128 = 7\)[/tex]
As we can see, option d) [tex]\(\log_2 128 = 7\)[/tex] matches our derived logarithmic equation perfectly. Therefore, the correct answer is:
d) [tex]\(\log_2 128 = 7\)[/tex]
Let's apply this to [tex]\(2^7 = 128\)[/tex]:
1. Base (a): The base here is 2.
2. Exponent (b): The exponent is 7.
3. Result (c): The result is 128.
Using the logarithmic form [tex]\(\log_a(c) = b\)[/tex], substitute the values we know:
[tex]\[
\log_2(128) = 7
\][/tex]
Now compare this with the given options:
- a) [tex]\(\log_7 2 = 128\)[/tex]
- b) [tex]\(\log_2 7 = 128\)[/tex]
- c) [tex]\(\log_7 128 = 2\)[/tex]
- d) [tex]\(\log_2 128 = 7\)[/tex]
As we can see, option d) [tex]\(\log_2 128 = 7\)[/tex] matches our derived logarithmic equation perfectly. Therefore, the correct answer is:
d) [tex]\(\log_2 128 = 7\)[/tex]
