Simplify the following expressions and solve the equations: a) e^ x2 b) 4 + 25 2a) 3 2 x + 2 x = 4 2b) 2e^-x + e^x = 3

In this case, $A = x^2$.
$$
e^{\ln x^2} = x^2
$$
The simplified expression is $\boxed{x^2}$.

• b) $\log 4 + \log 25$

(Assuming base 10 for $\log$ when no base is specified). $\log_{10} 100 = 2$ The simplified expression is $\boxed{2}$.

2) Solve for the equation

• a) $3 \log_2 x + \log_2 x = 4$

$$
x = 2^1
$$
$$
x = 2
$$
The solution is $\boxed{x=2}$.

• b) $2e^{-x} + e^x = 3$

Substitute $y$ into the equation. $\frac{2}{y} + y = 3$

For $y=1$:
$$
e^x = 1
$$
$$
x = \ln 1
$$
$$
x = 0
$$
For $y=2$:
$$
e^x = 2
$$
$$
x = \ln 2
$$
The solutions are $\boxed{x=0, \ln 2}$.

3) Given that $(x-2)$, $(x+1)$, and $(7x-2)$ are the first 3 consecutive terms of an AP.

• a) Find the value of $x$ and the common difference.

Therefore, we can equate the differences between the terms. $(x+1) - (x-2) = (7x-2) - (x+1)$

$$
d = (x+1) - (x-2)
$$
$$
d = (1+1) - (1-2)
$$
$$
d = 2 - (-1)
$$
$$
d = 3
$$
The value of $x$ is $\boxed{1}$ and the common difference is $\boxed{3}$.

• b) Find the range of values of $x$. In this problem, $x$ is a specific value that makes the given terms an arithmetic progression. There is no range of values for $x$. The value of $x$ is the single value found in part (a). The value of $x$ is $\boxed{1}$.