Sure, let's write the logarithmic equation [tex]\(\log_8 \frac{1}{512} = -3\)[/tex] in its exponential form and confirm its accuracy step by step.
Given:
[tex]\[
\log_8 \frac{1}{512} = -3
\][/tex]
The logarithmic equation [tex]\(\log_b y = x\)[/tex] implies the exponential form [tex]\(b^x = y\)[/tex].
Here:
- [tex]\(b\)[/tex] (the base of the logarithm) is [tex]\(8\)[/tex],
- [tex]\(y\)[/tex] (the argument of the logarithm) is [tex]\(\frac{1}{512}\)[/tex],
- [tex]\(x\)[/tex] (the result of the logarithm) is [tex]\(-3\)[/tex].
So, we can rewrite [tex]\(\log_8 \frac{1}{512} = -3\)[/tex] in exponential form as:
[tex]\[
8^{-3} = \frac{1}{512}
\][/tex]
Let’s check the equivalence by calculating [tex]\(8^{-3}\)[/tex]:
[tex]\[
8^{-3} = \frac{1}{8^3} = \frac{1}{512}
\][/tex]
Thus, the exponential form [tex]\(8^{-3} = \frac{1}{512}\)[/tex] is correct.
To further explore the equation and its transformations:
1. [tex]\(\left(\frac{1}{512}\right)^{-3}\)[/tex]:
[tex]\(\left(\frac{1}{512}\right)^{-3}\)[/tex] means taking the reciprocal of [tex]\(\frac{1}{512}\)[/tex] and raising it to the power of 3:
[tex]\[
\left(\frac{1}{512}\right)^{-3} = 512^3 = 8
\][/tex]
This step doesn't match our original requirement which started with [tex]\(8^{-3} = \frac{1}{512}\)[/tex].
2. [tex]\(\frac{1}{512} = 3^{-8}\)[/tex]:
[tex]\(\frac{1}{512}\)[/tex] being equal to [tex]\(3^{-8}\)[/tex] doesn't hold in our context since [tex]\(3^8\)[/tex] is not equal to [tex]\(512\)[/tex]:
[tex]\[
3^8 = 6561 \quad \text{not} \quad 512.
\][/tex]
Therefore, this is incorrect.
3. [tex]\(\frac{1}{512} = -3^8\)[/tex]:
This expression would imply taking the negative of the exponential of 3 raised to 8, which is not aligned with the positive values used in original logarithmic or exponential forms for [tex]\(8\)[/tex].
Thus, the only correct exponential form of [tex]\(\log_8 \frac{1}{512} = -3\)[/tex] is:
[tex]\[
8^{-3} = \frac{1}{512}
\][/tex]
This confirms and aligns correctly with both the logarithmic definition and our initial steps.
