To determine which logarithmic equation is equivalent to the exponential equation [tex]\(8^2 = 64\)[/tex], we'll convert the exponential form [tex]\(a^b = c\)[/tex] into the corresponding logarithmic form [tex]\(b = \log_a(c)\)[/tex].
Given:
[tex]\[ 8^2 = 64 \][/tex]
Here's a step-by-step solution:
1. Identify the base ([tex]\(a\)[/tex]), exponent ([tex]\(b\)[/tex]), and result ([tex]\(c\)[/tex]) from the exponential equation:
- Base ([tex]\(a\)[/tex]) = 8
- Exponent ([tex]\(b\)[/tex]) = 2
- Result ([tex]\(c\)[/tex]) = 64
2. The general conversion from exponential form [tex]\(a^b = c\)[/tex] to logarithmic form is:
[tex]\[ b = \log_a(c) \][/tex]
3. Substitute the identified values into the logarithmic form equation:
- [tex]\(a = 8\)[/tex]
- [tex]\(b = 2\)[/tex]
- [tex]\(c = 64\)[/tex]
This gives us:
[tex]\[ 2 = \log_8(64) \][/tex]
4. Compare [tex]\(2 = \log_8(64)\)[/tex] with the given options:
- [tex]\(2 = \log _8 64\)[/tex]
- [tex]\(2 = \log _{64} 8\)[/tex]
- [tex]\(8 = \log _2 64\)[/tex]
- [tex]\(64 = \log _2 8\)[/tex]
From these options, the one that matches our derived logarithmic equation is:
[tex]\[ 2 = \log_8(64) \][/tex]
Therefore, the correct equivalent logarithmic equation is:
[tex]\[ 2 = \log_8(64) \][/tex]
