The property of exponents states that when multiplying like bases, you add the exponents: m + n = p.

If a = b, then a^x = b^y implies x must equal y for the equality to hold.

Substituting x = 1 into the function gives f(1) = 3(1)^2 + 2(1) - 5 = 3 + 2 - 5 = 0.

Since 8 is 2^3, log_2(8) equals 3.

log_2(x) + 2 = 5 implies log_2(x) = 3, thus x = 2^3 = 8.

Since 27 is 3^3, log_3(27) = 3.

The expression log_a(b) = c can be rewritten in exponential form as a^c = b.

The definition of logarithms states that if log_a(b) = c, then a raised to the power of c equals b, hence a^c = b.

By the definition of logarithms, if log_a(b) = x, then b = a^x.

The definition of logarithms states that if log_b(x) = y, then b raised to the power of y equals x, hence b^y = x.

The sum of the first n terms S_n = n/2 * (2a + (n-1)d). Here, S_10 = 10/2 * (2*4 + 9*3) = 5 * (8 + 27) = 5 * 35 = 175.

Let the first term be a and the common ratio be r. Then, 2nd term = ar = 8 and 4th term = ar^3 = 32. Dividing these gives r^2 = 4, so r = 2.

Let the first term be a and the common ratio be r. Then, 2nd term = ar = 12 and 4th term = ar^3 = 48. Dividing these gives r^2 = 4, so r = 2.

Let the first term be a and the common ratio be r. Then, 2nd term = ar = 8 and 4th term = ar^3 = 32. Dividing gives r^2 = 4, so r = 2.

Let the first term be a and the common difference be d. From the equations a + d = 10 and a + 4d = 16, we can solve for d to find it is 2.

Let the first term be a and the common difference be d. From a + d = 10 and a + 4d = 16, we can find a + 2d = 12, which is the 3rd term.

Let the first term be a and the common difference be d. From the equations a + d = 15 and a + 3d = 25, we can find d = 5.

Let the first term be a and the common difference be d. From the equations a + d = 8 and a + 3d = 14, we can find a = 6.

Let the first term be a and the common difference be d. From the equations a + d = 8 and a + 4d = 14, we can find the 3rd term a + 2d = 10.

Let the first term be a and the common difference be d. From the equations a + d = 8 and a + 4d = 20, we can solve for a to find it equals 4.

Let the first term be a. Then, the 3rd term is ar^2 = 27. Thus, a * 3^2 = 27, giving a = 3.

Let the first term be a and the common difference be d. We have a + 2d = 12 and a + 6d = 24. Subtracting these gives 4d = 12, so d = 3.

Let the first term be a and the common difference be d. From the equations a + 2d = 15 and a + 5d = 24, solving gives d = 3.

Let the first term be a and the common difference be d. From the equations a + 2d = 15 and a + 6d = 27, solving gives d = 3.

Let the first term be a and the common difference be d. From the equations a + 4d = 15 and a + 9d = 30, we can find d = 3.

Let the first term be a and the common difference be d. From the equations a + 4d = 20 and a + 9d = 35, we can solve for a to find it is 10.

Let the first term be a and the common difference be d. From the equations a + 5d = 30 and a + 8d = 45, we can find d = 5.

Using the formula for the nth term, a + 6d = 25. Substituting d = 3 gives a + 18 = 25, thus a = 7.

Using the formula for the nth term, a + 6d = 50. Substituting d = 5 gives a + 30 = 50, hence a = 20.

In a GP with a negative common ratio, the terms will alternate in sign, starting from the sign of the first term.