Which logarithmic equation is equivalent to [tex]$8^2=64?$[/tex]

A. [tex]2 = \log _8 64[/tex]
B. [tex]2 = \log _{64} 8[/tex]
C. [tex]8 = \log _2 64[/tex]
D. [tex]64 = \log _2 8[/tex]



Answer :

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]