Solve for x: log5(x-2)+ log5 (x-4) = log6(x-1)
log5 ( (x-2)(x-4) ) = log6 (x-1)
clearly x > 4
log (x^2 - 6x + 8) /log5 = log(x-1)/log6
log6 log(x^2 - 6x + 8) = log5 log(x-1)
log (x^2 - 6x + 8)^(log6) = log (x-1)^(log5)
anti-log it
(x^2 - 6x + 8)^log6 = (x-1)^log5
..... oh my, did you really mean for those bases to be diffferent ??
I ran it through Wolfram and got
x = 5.1409 , rejecting the smaller.
http://www.wolframalpha.com/input/?i=log%28+%28x-2%29%28x-4%29+%29%2Flog%285%29+%3D+log%28x-1%29%2Flog%286%29
To solve the equation log5(x-2) + log5(x-4) = log6(x-1), we need to use logarithmic properties and algebraic steps. Here's the step-by-step solution:
Step 1: Combine the logarithms using the product rule:
log5[(x-2)(x-4)] = log6(x-1)
Step 2: Apply the change of base formula to convert the logarithm from base 6 to base 5:
log5[(x-2)(x-4)] = log6(x-1) / log6(5)
Step 3: Simplify the right side using the change of base formula:
log5[(x-2)(x-4)] = log5(x-1) / log5(6)
Step 4: Apply the quotient rule:
log5[(x-2)(x-4)] = log5(x-1) - log5(6)
Step 5: Use the logarithmic identity logb(mn) = logb(m) + logb(n) to rewrite the left side:
log5[(x-2)(x-4)] = log5(x-1) - log5(6)
Step 6: Apply the exponentiation property to get rid of the logarithms:
[(x-2)(x-4)] = (x-1) / 6
Step 7: Expand and simplify the equation:
x^2 - 6x + 8 = (x-1) / 6
Step 8: Multiply both sides of the equation by 6 to eliminate the denominator:
6(x^2 - 6x + 8) = x - 1
Step 9: Distribute and simplify:
6x^2 - 36x + 48 = x - 1
Step 10: Rearrange and set the equation to zero:
6x^2 - 37x + 49 = 0
Step 11: Solve the quadratic equation using factoring, completing the square, or the quadratic formula. In this case, we'll use the quadratic formula:
x = [-(-37) ± √((-37)^2 - 4(6)(49))] / (2(6))
Step 12: Simplify the expression under the square root:
x = [37 ± √(1369 - 1176)] / 12
Step 13: Continue simplifying:
x = [37 ± √193] / 12
Therefore, the solutions for x are:
x = (37 + √193) / 12
x = (37 - √193) / 12
To solve the given equation:
log5(x-2) + log5(x-4) = log6(x-1)
We can use the properties of logarithms to rewrite the equation. The log property states that the sum of logarithms with the same base is equal to the logarithm of the product of the numbers.
Using this property, we can combine the two logarithms on the left side of the equation:
log5[(x-2)(x-4)] = log6(x-1)
Now, we have a single logarithm on both sides of the equation with the same base. Therefore, the arguments inside the logarithms must be equal:
(x-2)(x-4) = x-1
Next, we can distribute and simplify the left side of the equation:
x^2 - 6x + 8 = x - 1
Now, we have a quadratic equation. To solve it, we bring all the terms to one side:
x^2 - 7x + 9 = 0
To factor or use other methods to solve this quadratic equation, it's not immediately apparent. Thus, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a, b, and c correspond to the coefficients in the quadratic equation (ax^2 + bx + c = 0). In our case:
a = 1, b = -7, c = 9
Substituting these values into the quadratic formula:
x = (-(-7) ± √((-7)^2 - 4(1)(9))) / (2(1))
Simplifying:
x = (7 ± √(49 - 36)) / 2
x = (7 ± √13) / 2
So, the solutions for x are:
x = (7 + √13) / 2
x = (7 - √13) / 2