Let's solve the equation \(\log_{256} (2x - 1) = 8\) step by step.
1. Understanding the Logarithmic Equation:
The equation given is \(\log_{256} (2x - 1) = 8\). This means that \(256\) raised to the power of \(8\) equals \(2x - 1\).
2. Converting the Logarithmic Equation to an Exponential Equation:
[tex]\[
256^8 = 2x - 1
\][/tex]
3. Simplifying the Exponential Expression:
Now we need to simplify \(256^8\).
Notice that \(256\) can be written as \(2^8\) because:
[tex]\[
256 = 2^8
\][/tex]
Therefore,
[tex]\[
256^8 = (2^8)^8
\][/tex]
4. Further Simplify Using Power Rules:
Using the power rule \((a^m)^n = a^{m \cdot n}\), we get:
[tex]\[
(2^8)^8 = 2^{64}
\][/tex]
5. Substitute Back:
So now we have:
[tex]\[
2x - 1 = 2^{64}
\][/tex]
6. Solving for \(x\):
To isolate \(x\), we add \(1\) to both sides of the equation:
[tex]\[
2x = 2^{64} + 1
\][/tex]
Then, divide by \(2\) to solve for \(x\):
[tex]\[
x = \frac{2^{64} + 1}{2}
\][/tex]
7. Simplifying the Expression:
This simplifies to:
[tex]\[
x = \frac{2^{64}}{2} + \frac{1}{2} = 2^{63} + \frac{1}{2}
\][/tex]
8. Final Answer:
As simplified, we have:
[tex]\[
x = \frac{257}{2}
\][/tex]
Therefore, the solution to the equation \(\log_{256} (2x - 1) = 8\) is:
[tex]\[
x = \frac{257}{2}
\][/tex]
