Solve log.
2 log (x-1)= log (x+1)
the answer is 3 but i don't know how to get it...Any one hint or suggestion will be greatly appreciated :)
2 log (x-1)= log (x+1)
log[(x-1)²] = log(x+1)
Take antilog:
(x-1)²=x+1
Expand and solve quadratic:
x²-3x=0
x(x-3)=0
x=0 is not admissible as solution because log(0-1) is undefined.
The remaining answer is therefore x=3.
To solve the given logarithmic equation, we'll make use of the properties of logarithms.
First, let's apply the power rule of logarithms, which states that for any positive numbers a, b, and n:
log base b of (a^n) = n * log base b of a
Now, let's simplify the equation step by step:
2log(x-1) = log(x+1)
Using the power rule, we can rewrite the left side of the equation:
log((x-1)^2) = log(x+1)
Now, we have the same base on both sides of the equation (log base 10), so we can eliminate the logarithms:
(x-1)^2 = x+1
Expanding the left side by squaring:
x^2 - 2x + 1 = x + 1
Rearranging the equation to bring all terms to one side:
x^2 - 3x = 0
Factoring out an x:
x(x - 3) = 0
Now, we set each factor equal to zero and solve for x:
x = 0 or x - 3 = 0
From the first equation, x = 0.
From the second equation, x = 3.
Therefore, the possible solutions to the equation are x = 0 and x = 3.
However, we have to verify if these solutions satisfy the original equation since we dealt with logarithms. Substituting both values into the original equation, we find that only x = 3 satisfies the equation.
Thus, the solution to the equation log(2(x-1)) = log(x+1) is x = 3.