a)

i)

We can rearrange logarithms using indices. 

$$\log_ba=c \Longleftrightarrow a=b^c$$

Using this equivalence we can rewrite $\log_\var{f}x=\var{f1}$.

$$\begin{align} x&= \var{f}^\var{f1} \\ &=\var{f^f1} \end{align}$$

b)

i)

We can use the equivalence to rewrite our equation.

$$\log_ba=c \Longleftrightarrow a=b^c$$

We can write out our values to makes it easier.

$$\begin{align} a&=x \\ b&=\var{g1}\\ c&=y+\var{g2} \end{align}$$

Then we can write out our equation in the required form.

$$x=\var{g1}^{y+\var{g2}}$$

c)

We can use the same equivalence as in part b). 

$$\log_ba=c \Longleftrightarrow a=b^c$$

We have

$$\begin{align} a&=y+\var{h1} \\ b&=x\\ c&=\var{h2}\text{.} \\ \\ \log_{x}(y+\var{h1}) &= \var{h2} \\ \implies y+\var{h1} &= x^{\var{h2}} \\ x &= (y+\var{h1})^{\frac{1}{\var{h2}}} \end{align}$$

d) 

The two in this list that don't equal $x$ are $\log_e(x)$ and $\log_{10}(x)$.

$$\begin{align} \log_e(x)&=\ln(x)\\ \log_{10}(x)&=\log(x)\text{.} \end{align}$$