Solve algebraically for x:

\[
\log_{27} (2x - 1) = \frac{4}{3}
\]

To solve the equation log base 27 (2x-1) = 4/3 algebraically, you need to use the definition of logarithms to convert the logarithmic equation to an exponential equation. Recall the property that log base b of a number is c if and only if b to the power of c equals that number. Using this property:
Rewrite the equation as an exponential equation: 27(4/3) = 2x - 1.
Solve for x: 2x = 27(4/3) + 1. To simplify, cube both sides of the equation to eliminate the fractional exponent: (33)(4/3) = 34 = 81.
Therefore, 2x = 81 + 1 which equals 82.
Now, divide both sides by 2 to get x = 82 / 2, which simplifies to x = 41.
So, the solution to the logarithmic equation is x = 41.

Answered by Qwletter | 2024-06-24

To solve lo g 27 ​ ( 2 x − 1 ) = 3 4 ​ , convert it to exponential form to find that x = 41 . This involves calculating 2 7 3 4 ​ and rearranging the equation accordingly. Therefore, the final answer is x = 41 .
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Answered by Qwletter | 2024-12-23