To solve this logarithmic equation, we can first convert both sides to exponential form:
5^(Log5(5x-1)) = 5^(Log5(1-x) + 1)
On the left side, the base 5 and the logarithm log5 cancel out, leaving us with just 5x-1:
5x - 1 = 5^(Log5(1-x)) * 5
Now, we need to solve for x. Let's simplify the right side by recognizing that log5(1-x) + 1 is equivalent to log5(1-x) + log5(5), which simplifies to log5(5(1-x)):
5x - 1 = 5^(Log5(5(1-x)))
Now, notice that 5^(Log5(5(1-x))) is equal to 5(1-x) because the base 5 and the logarithm log5 cancel out:
5x - 1 = 5(1-x)
Simplify both sides:
5x - 1 = 5 - 5x
Now, solve for x:
5x + 5x = 5 + 1 10x = 6 x = 6/10 x = 3/5
Therefore, the solution to the equation log5(5x-1) = log5(1-x) + 1 is x = 3/5.
To solve this logarithmic equation, we can first convert both sides to exponential form:
5^(Log5(5x-1)) = 5^(Log5(1-x) + 1)
On the left side, the base 5 and the logarithm log5 cancel out, leaving us with just 5x-1:
5x - 1 = 5^(Log5(1-x)) * 5
Now, we need to solve for x. Let's simplify the right side by recognizing that log5(1-x) + 1 is equivalent to log5(1-x) + log5(5), which simplifies to log5(5(1-x)):
5x - 1 = 5^(Log5(5(1-x)))
Now, notice that 5^(Log5(5(1-x))) is equal to 5(1-x) because the base 5 and the logarithm log5 cancel out:
5x - 1 = 5(1-x)
Simplify both sides:
5x - 1 = 5 - 5x
Now, solve for x:
5x + 5x = 5 + 1
10x = 6
x = 6/10
x = 3/5
Therefore, the solution to the equation log5(5x-1) = log5(1-x) + 1 is x = 3/5.