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).