a) $\displaystyle 25^{x-3}=5^x$
You can get the same base here since $\displaystyle 25=5^2$, so that you may write:
$\displaystyle (5^2)^{x-3}=5^x$
$\displaystyle 5^{2(x-3)}=5^x$
Now equate exponents, and solve for $\displaystyle x$.
b) $\displaystyle 2\left(\frac{1}{2} \right)^x-\left(\frac{1}{4} \right)^x=1$
Use the substitution $\displaystyle u=\left(\frac{1}{2} \right)^x$ so that you may nor write:
$\displaystyle 2u-u^2=1$
Now, solve the resulting quadratic, then back substitute for $\displaystyle u$.
c) $\displaystyle 3\ln(2x+5)=9$
Divide through by 3 to get:
$\displaystyle \ln(2x+5)=3$
Now convert from logarithmic to exponential form and solve for $\displaystyle x$.
d) $\displaystyle e^{(x-2)^2}=2$
Convert from exponential to logarithmic form:
$\displaystyle (x-2)^2=\ln(2)$
Now solve for $\displaystyle x$ using the square root property.
e) $\displaystyle 6^{-7x}=5$
Take the natural log of both sides to get:
$\displaystyle -7x\ln(6)=\ln(5)$
Now solve for $\displaystyle x$.