Solve for x:
1.) ln√x-8=5
2.) log2x+log2(x+2)=log2(x+6)
ln√x-8=5
ln√x=13
√x = e^13
x = e^26
log2x+log2(x+2)=log2(x+6)
log2(x(x+2)) = log2(x+6)
x(x+2) = x+6
x^2+2x = x+6
x^2+x-6 = 0
(x-2)(x+3) = 0
x = 2 or -3
But, log2(x) is not defined for x <= 0, so
x=2 is the only solution
check:
log2(2)+log2(2+2) = log2(2+6)
1+2 = 3
To solve for x in these equations, we'll go through each equation step by step.
1.) ln√x - 8 = 5
Step 1: Add 8 to both sides of the equation:
ln√x = 13
Step 2: Eliminate the natural logarithm by expressing it as an exponential function:
e^(ln√x) = e^13
√x = e^13
Step 3: Square both sides of the equation to eliminate the square root:
(√x)^2 = (e^13)^2
x = e^26
So the solution to this equation is x = e^26.
2.) log2x + log2(x + 2) = log2(x + 6)
Step 1: Combine the logarithms using logarithmic properties:
log2(x(x + 2)) = log2(x + 6)
Step 2: Set the inside of the logarithms equal to each other:
x(x + 2) = x + 6
Step 3: Expand and rearrange the equation to simplify it:
x^2 + 2x = x + 6
x^2 + x - 6 = 0
Step 4: Factor or use the quadratic formula to solve for x:
(x + 3)(x - 2) = 0
Setting each factor equal to zero:
x + 3 = 0 or x - 2 = 0
Solving for x in each case:
x = -3 or x = 2
So the solutions to this equation are x = -3 and x = 2.