1. if P(t)=3e^2t, for what value does P(t)= 2P(2)
2. Find all x for which the square root of log (x)= log square root of x
thanks!!
1. If P(t) =3P(2)
then 3e^(2t) = 2(3e^(4t)
e^(2t) = 2 (e^2t)^2
let e^2t = y
y = 2y^2
2y^2 - y = 0
y(2y -1) = 0
y = 0 or y = 1/2
e^2t = 0 ---> no solution
or
e^2t = 1/2
ln (e^2t) = ln (1/2) = ln1 - ln2 = -ln2
2t = -ln2
t = (-1/2)ln 2
2. √(log(x)) = log (√x) = log x^(1/2) = (1/2) logx
let logx = y
√ y = (1/2)y
y = (1/4)y^2
4y = y^2
y^2 - 4y = 0
y (y - 4) = 0
y = 0 or y = 4
logx = 0 or logx = 4
x = 10^0 = 1 OR x = 10^4 = 10000
My error for #1, see "bobpursley's" answer
http://www.jiskha.com/display.cgi?id=1355365995
1. To find the value of t for which P(t) is equal to 2P(2), we need to set up an equation and solve for t.
Given that P(t) = 3e^(2t), we can substitute P(2) into the equation:
2P(2) = 2 * 3e^(2(2))
Simplifying:
2P(2) = 6e^4
Now we can set up the equation:
3e^(2t) = 6e^4
Dividing both sides by 3:
e^(2t) = 2e^4
Taking the natural logarithm (ln) of both sides:
ln(e^(2t)) = ln(2e^4)
Using the logarithmic property of exponents:
2t * ln(e) = ln(2) + ln(e^4)
Since ln(e) = 1:
2t = ln(2) + 4
Dividing by 2:
t = (ln(2) + 4) / 2
Calculating the value of t:
t = (0.693 + 4) / 2 ≈ 4.3465
So, the value of t for which P(t) = 2P(2) is approximately 4.3465.
2. To find all x for which the square root of log(x) is equal to the logarithm of the square root of x, we can set up an equation and solve for x.
Given that √log(x) = log(√x), we can square both sides of the equation to eliminate the square root:
(√log(x))^2 = (log(√x))^2
Simplifying:
log(x) = log^2(√x)
Using the logarithmic property of exponents, we can rewrite the equation:
x = (√x)^log^2(√x)
Taking the square of both sides:
x^2 = (√x)^(2 * log^2(√x))
Using the logarithmic property again to simplify the right side:
x^2 = (√x)^(log^2(x))
Since (√x)^(log^2(x)) is equivalent to x^(log^2(x)/2), we can rewrite the equation:
x^2 = x^(log^2(x)/2)
Now we can equate the exponents:
2 = log^2(x)/2
Taking the square root of both sides:
√2 = log(x)/2
Multiplying both sides by 2:
2√2 = log(x)
Now we can rewrite this equation in exponential form:
x = 10^(2√2)
So, all x for which the square root of log(x) is equal to the logarithm of the square root of x are given by x = 10^(2√2).