Find the exact solution, using common logarithms, and a two-decimal-place approximation of each solution.

log(7x + 4) = 2 + log(2x − 3)

Solve the equation.

log(x^3) = (log x)^2

log(7x + 4) = 2 + log(2x − 3)

log(7x + 4) - log(2x − 3) = 2
log( (7x+4)/(2x-3) ) = 2
(7x+4)/(2x-3) = 10^2 = 100
200x - 300 = 7x+4
193x = 304
x = 304/193 or appr 1.575

for the 2nd

log x^3 = log x^2
x^3 = x^2
x^3 - x^2 = 0
x^2(x-1) = 0
x = 0 or x = 1

but from log(2x-3) , 2x-3 > 0
2x > 3
x > 3/2

so there is not solution.

Well, solving equations can be quite logarithm-ic, but I'll do my best to give you some clownish solutions!

For the first equation, let's use some clown logarithmic magic. We can rewrite the equation as 2 + log(2x - 3) = log(7x + 4).

Now, let's combine the logarithms on the right side: log((2x - 3)(7x + 4)) = 2.

Next, we can use the property of logarithms to rewrite the equation as (2x - 3)(7x + 4) = 10^2.

Expanding the equation, we get 14x^2 - 5x - 12 = 100.

Now, let's bring everything to one side and set it equal to zero: 14x^2 - 5x - 112 = 0.

Using the quadratic formula, we can solve for x. The two solutions are x = 4.57 and x = -3.34 (rounded to two decimal places).

Moving on to the second equation, we have log(x^3) = (log x)^2.

By changing the logarithm on the right side to an exponent form, we get log(x^3) = log^2(x).

Now, we can take the exponent form and equate the exponents: x^3 = x^2.

Dividing both sides by x^2, we get x = 1.

So, the exact solution for the second equation is x = 1, and the two-decimal-place approximation is x ≈ 1.00.

Remember, I'm just a clown bot, so don't clown on me too hard if my answers aren't perfect!

To solve the equation log(7x + 4) = 2 + log(2x - 3), we can use the properties of logarithms.

Step 1: Combine the logarithms on the right side of the equation.
log(7x + 4) = log(10^(2)) + log(2x - 3)

Step 2: Simplify the logarithm on the right side.
log(7x + 4) = log(100) + log(2x - 3)
log(7x + 4) = log(100(2x - 3))

Step 3: Set the expressions inside the logarithms equal to each other.
7x + 4 = 100(2x - 3)

Step 4: Expand and solve the equation.
7x + 4 = 200x - 300
4 + 300 = 200x - 7x
304 = 193x
x = 304/193

To find the two-decimal-place approximation of the solution, divide 304 by 193.
x ≈ 1.57 (rounded to two decimal places)

Now, let's solve the equation log(x^3) = (log x)^2 using common logarithms.

Step 1: Apply the power rule of logarithms to the left side of the equation.
3log(x) = (log x)^2

Step 2: Let y = log(x).
3y = y^2

Step 3: Rearrange the equation.
y^2 - 3y = 0

Step 4: Factorize the equation.
y(y - 3) = 0

Step 5: Set each factor equal to zero.
y = 0 or y - 3 = 0

Step 6: Solve for y.
y = 0 or y = 3

Step 7: Substitute back y = log(x).
log(x) = 0 or log(x) = 3

For log(x) = 0, we know that 10^0 = x, so x = 1.

For log(x) = 3, we know that 10^3 = x, so x = 1000.

Therefore, the two solutions to the equation log(x^3) = (log x)^2 using common logarithms are x = 1 and x = 1000.

To solve the equation log(7x + 4) = 2 + log(2x − 3), we need to use the properties of logarithms.

Step 1: Combine the logarithms on the right side of the equation.

log(7x + 4) = log[(2x - 3) * 10^2] (using the property log(a) + log(b) = log(ab) and 2 = log(100))

Step 2: Apply the property log(a^b) = b * log(a) to simplify.

log(7x + 4) = log[(2x - 3) * 100]

Step 3: Since the logarithms on both sides have the same base (which is 10 in this case), we can equate the expressions inside the logarithms.

7x + 4 = (2x - 3) * 100

Step 4: Simplify the equation.

7x + 4 = 200x - 300

Step 5: Move all the x terms to one side and the constant terms to the other side.

7x - 200x = -300 - 4

Step 6: Combine like terms.

-193x = -304

Step 7: Solve for x.

x = -304 / -193

Using a calculator to evaluate this expression, we get x ≈ 1.577.

To find the two-decimal-place approximation, the solution is x ≈ 1.58.

Now, let's solve the equation log(x^3) = (log x)^2 using common logarithms.

Step 1: Apply the power rule for logarithms.

3 * log x = (log x)^2

Step 2: Rearrange the equation to obtain a quadratic form.

(log x)^2 - 3 * log x = 0

Step 3: Factor out log x.

log x * ((log x) - 3) = 0

Step 4: Set each factor equal to 0 and solve for x.

log x = 0 or log x - 3 = 0

Step 5: Solve each equation separately.

For log x = 0, x = 10^0 = 1.

For log x - 3 = 0, log x = 3, then x = 10^3 = 1000.

Therefore, the exact solutions to the equation log(x^3) = (log x)^2 using common logarithms are x = 1 and x = 1000.

The two-decimal-place approximations are x ≈ 1.00 and x ≈ 1000.00.