a) To solve the equation 4.2^x - 3 = 1, we can start by adding 3 to both sides of the equation:

4.2^x = 4

Next, we can take the logarithm of both sides of the equation to eliminate the exponent:

log(4.2^x) = log(4)

Using the logarithmic property log(a^b) = b*log(a), we can bring down the exponent:

x*log(4.2) = log(4)

Now, we can divide both sides of the equation by log(4.2) to solve for x:

x = log(4) / log(4.2)

Using a calculator, we find that x ≈ -0.096.

b) To solve the equation 3^x+2 - 5.3^x = 36, we can start by subtracting 36 from both sides of the equation:

3^x+2 - 5.3^x - 36 = 0

Next, we can simplify the equation by combining like terms:

3^x+2 - 5.3^x = 36

3^x * 3^2 - 5 * 3^x * 3^x = 36

9 * 3^x - 5 * (3^x)^2 = 36

Now, let's substitute a variable to simplify the equation. Let's say u = 3^x:

9u - 5u^2 = 36

Rearranging the equation, we get:

5u^2 - 9u + 36 = 0

This is a quadratic equation in terms of u. We can solve it using factoring, completing the square, or the quadratic formula. However, upon inspection, we can see that there are no real solutions for u. Therefore, there are no real solutions for x in this equation.

c) To solve the equation 7.4^x-1 + 4^x+1 = 23, we can start by simplifying the equation:

7.4^x * 7.4^-1 + 4^x * 4^1 = 23

7.4^x / 7.4 + 4^x * 4 = 23

Now, let's substitute a variable to simplify the equation. Let's say u = 7.4^x:

u / 7.4 + 4^x * 4 = 23

Multiplying both sides of the equation by 7.4, we get:

u + 4^x * 4 * 7.4 = 23 * 7.4

u + 4^x * 29.6 = 170.2

Substituting back u = 7.4^x, we get:

7.4^x + 4^x * 29.6 = 170.2

Now, we have a quadratic equation in terms of 4^x. We can solve it using factoring, completing the square, or the quadratic formula. However, this equation does not have a simple solution. We would need to use numerical methods or a calculator to approximate the value of x.