solve the logarithmic equation . express solution in exact form
log5(x-9)+log5(x+4)=1+log5(x-5)
1 = log5(5), so we have
log5(x-9)+log5(x+4)=log5(5)+log5(x-5)
log5[(x-9)(x+4)] = log5[5(x-5)]
raise 5 to the powers, and we have
(x-9)(x+4) = 5(x-5)
x^2 - 5x - 36 = 5x - 25
x^2 - 10x - 11 = 0
(x-11)(x+1) = 0
Solutions are 11,-1
However, -1 does not fit the original equation: log of negatives are undefined.
log5(x-9)+log5(x+4)=1+log5(x-5)
log5(x-9)+log5(x+4)=log5(5)+log5(x-5)
log5[(x-9)(x+4)] = log5[5(x-5)}
(x-9)(x+4) = 5(x-5)
x^2 - 5x - 36 = 5x - 25
x^2 - 10x - 9 = 0
x = (10 ± √136)/2 = appr. 10.83 or -.83
but for each of the above to defined, x > 9
so x = (10 + √136)/2 = 5 + √34
check my arithmetic
I have an error in my equation...
x^2 - 10x - 9 = 0 should be
x^2 - 10x - 11 - 0 , just like Steve had
then (x-11)(x+1) = 0
x = 11 or x = -1
so x = 11
To solve the logarithmic equation log5(x-9) + log5(x+4) = 1 + log5(x-5), we will use the properties of logarithms to simplify the equation.
Step 1: Combine the logarithms using the product rule.
log5[(x-9)(x+4)] = log5[(x-5)] + 1
Step 2: Simplify the equation further.
log5[(x^2 - 5x - 36)] = log5[(x-5) * 5]
Step 3: Apply the power rule to eliminate the logarithm.
x^2 - 5x - 36 = (x - 5) * 5
Step 4: Expand and rearrange the equation.
x^2 - 5x - 36 = 5x - 25
Step 5: Simplify the equation by combining like terms.
x^2 - 10x + 11 = 0
Step 6: Solve the quadratic equation.
We can use the quadratic formula to find the exact form of the solutions.
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -10, and c = 11. Plugging these values into the quadratic formula, we get:
x = (-(-10) ± √((-10)^2 - 4(1)(11))) / (2(1))
= (10 ± √(100 - 44)) / 2
= (10 ± √56) / 2
= 5 ± √14
So, the solutions to the logarithmic equation log5(x-9) + log5(x+4) = 1 + log5(x-5) in exact form are x = 5 + √14 and x = 5 - √14.