Solve for \( x \):

[tex]\[ \log (2x - 1) 256 = 8 \][/tex]

(Note: It appears there is a formatting or syntax issue in the original LaTeX code. If the intention is to solve for [tex]\( x \)[/tex] in the equation [tex]\( \log_{2x - 1} 256 = 8 \)[/tex], this corrected formatting should be used. If it is intended to be a different type of logarithmic equation, please specify.)



Answer :

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]