1.Express (xe^x)^2=30e^(-x) in the form lnx=ax+b,and find the value of a and of b .

2.solve 4(3^2x)=e^x

1. take ln of both sides

ln(x e^x)^2 = ln(30 e^-x)
2 lnx + 2ln e^x = ln30 + ln e^-x
2lnx + 2x = ln30 -x
2 lnx = -3x + ln30
lnx = (-3/2)x + (1/2)ln30 or (-3/2)x + ln (√30)
comparing this with ax + b
a = -3/2 , b = ln √30

2. take ln of both sides and use rules of logs

ln (4(3^2x) ) = ln e^x
ln4 + xln3 = x
x - 2xln3 = ln4
x(1 - 2ln3) = ln4
x = ln4/(1-2ln3) or appr. -1.1579

check:
LS = 4(3^-2.3158... = .31413784
RS = e^-1.157923.. = .31413784 , how about that?

1. Well, this equation seems to have a bad case of "math fever". Let's try to cure it with some humor medicine!

Now, let's tackle the equation (xe^x)^2 = 30e^(-x).

First, let's simplify it a bit: (xe^x)(xe^x) = 30e^(-x).

Now, we can use the properties of exponents. Adding the exponents on the left side gives us x^2e^(2x) = 30e^(-x).

Taking the natural logarithm (ln) of both sides should help us bring out some characters. Let's do it!

ln(x^2e^(2x)) = ln(30e^(-x)).

Using some log rules and splitting the terms, we have: ln(x^2) + ln(e^(2x)) = ln(30) + ln(e^(-x)).

Simplifying further, we have: 2ln(x) + 2x = ln(30) - x.

Since the equation is now in the form lnx=ax+b, we can identify a=2 and b=ln(30).

2. Let's have some fun and solve the equation 4(3^(2x)) = e^x, which seems to be in a slightly better mood!

To solve this, let's balance the equation using some math antics!

Dividing both sides by 4, we get: 3^(2x) = (1/4)e^x.

Now, let's bring out the big guns (ln) to solve this equation!

Taking the natural logarithm (ln) of both sides, we have: ln(3^(2x)) = ln((1/4)e^x).

Using the power rule of logarithms, we can rewrite the equation as: 2x ln(3) = ln(1/4) + x.

Subtracting x from both sides, we get: 2x ln(3) - x = ln(1/4).

Now, let's factor out the x from the left side: x(2 ln(3) - 1) = ln(1/4).

Finally, we isolate x by dividing both sides by (2 ln(3) - 1): x = ln(1/4) / (2 ln(3) - 1).

And there you have it! The solution to the equation 4(3^(2x)) = e^x is x = ln(1/4) / (2 ln(3) - 1). Now go celebrate with some mathematical humor!

1. To express the equation (xe^x)^2 = 30e^(-x) in the form lnx = ax + b, we need to take the natural logarithm (ln) on both sides of the equation.

ln[(xe^x)^2] = ln(30e^(-x))

Using logarithm properties, we can simplify the left side of the equation:

2ln(xe^x) = ln(30) + ln(e^(-x))

Next, we can use the logarithmic rule that ln(ab) = ln(a) + ln(b) to simplify further:

2[ln(x) + ln(e^x)] = ln(30) + (-x)

Simplifying the equation gives:

2[ln(x) + x] = ln(30) - x

Now, we have the equation in the desired form, lnx = ax + b:

ln(x^2) + 2x = ln(30) - x

2x + x = ln(30) - ln(x^2)

3x = ln(30) - 2ln(x)

Now, we can extract the value of a and b:

a = 3
b = ln(30) - 2ln(x)

2. To solve the equation 4(3^(2x)) = e^x, we can solve it step-by-step:

First, let's simplify the equation by applying exponent rules:

4(9^x) = e^x

Next, we can rewrite 4 as 2^2 and 9 as 3^2:

(2^2)(3^2x) = e^x

Applying the exponent rule that (a^m)(a^n) = a^(m+n):

2^(2+2x) * 3^(2x) = e^x

Using the exponent rule that a^(bc) = (a^b)^c:

2^2 * (2^x)^2 * 3^(2x) = e^x

Simplifying:

4 * (2^x)^2 * 3^(2x) = e^x

Now, the equation is in a quadratic form. We can rewrite (2^x)^2 as (2^x)(2^x) = 2^(2x):

4 * 2^(2x) * 3^(2x) = e^x

Applying the exponent rule that a^b * a^c = a^(b+c):

4 * (2 * 3)^(2x) = e^x

Using the property of exponents that (a^b)^c = a^(b*c):

4 * 6^(2x) = e^x

Now, we can see that we have the variable x in both the base and exponent. We can take the natural logarithm (ln) on both sides to solve for x:

ln(4 * 6^(2x)) = ln(e^x)

Using the logarithm property ln(ab) = ln(a) + ln(b):

ln(4) + ln(6^(2x)) = x

Using the exponent rule ln(a^b) = b * ln(a):

ln(4) + 2x * ln(6) = x

Subtracting x from both sides to get all the x terms on one side:

2x * ln(6) - x = -ln(4)

Factoring out x:

x(2ln(6) - 1) = -ln(4)

Dividing both sides by (2ln(6) - 1):

x = -ln(4) / (2ln(6) - 1)

This is the solution for x.

1. To express the given equation in the form lnx = ax + b, we can take the natural logarithm (ln) of both sides of the equation. Here's how to do it step-by-step:

Given: (xe^x)^2 = 30e^(-x)

Step 1: Take the natural logarithm (ln) of both sides:
ln((xe^x)^2) = ln(30e^(-x))

Step 2: Apply the power rule of logarithms to the left side:
2ln(xe^x) = ln(30e^(-x))

Step 3: Apply the product rule of logarithms to the left side:
ln(x) + ln(e^x) + ln(e^x) = ln(30e^(-x))

Step 4: Simplify using the properties of logarithms:
ln(x) + x + x = ln(30) - x

Step 5: Combine like terms:
2x + ln(x) = ln(30) - x

Step 6: Move all terms involving x to one side of the equation:
2x + x + ln(x) + x = ln(30)

Step 7: Simplify:
4x + ln(x) = ln(30)

Now we have the equation in the desired form lnx = ax + b, where a = 4 and b = ln(30).

2. To solve the equation 4(3^2x) = e^x, we need to isolate the variable x. Here's how to do it step-by-step:

Given: 4(3^2x) = e^x

Step 1: Simplify the exponential expression on the left side:
4(9^x) = e^x

Step 2: Write both sides of the equation with the same base (e):
(2^2)(9^x) = e^x

Step 3: Rewrite 9^x as (3^2)^x and apply the exponent rule (a^b)^c = a^(b*c):
2^2 * (3^2)^x = e^x

Step 4: Apply the power rule of exponents to the left side:
2^2 * 3^(2x) = e^x

Step 5: Rewrite 2^2 as 4:
4 * 3^(2x) = e^x

Step 6: Divide both sides of the equation by e^x:
4 * 3^(2x) / e^x = 1

Step 7: Rewrite 3^(2x) / e^x as (3^2 / e)^x:
4 * (3^2 / e)^x = 1

Step 8: Rewrite 3^2 / e as 9 / e:
4 * (9 / e)^x = 1

Step 9: Since any non-zero number raised to the power of zero is equal to 1, we can conclude that (9 / e)^x = 1 / 4.

Step 10: Take the logarithm (ln) of both sides to solve for x:
ln((9 / e)^x) = ln(1 / 4)

Step 11: Apply the logarithmic property to bring down the exponent:
x * ln(9 / e) = ln(1 / 4)

Step 12: Divide both sides by ln(9 / e) to solve for x:
x = ln(1 / 4) / ln(9 / e)

Using a calculator to evaluate the right side of the equation, we can find the numerical value of x.