How should I deduce the variance and expectation of the log of a variable? I read this paper "voom: precision weights unlock linear model analysis tools for RNA-seq read counts", in the methods, the "Delta rule for log-cpm" section:
The RNA-seq data consist of a matrix of read counts $r_{gi}$, for RNA samples $i=1$ to $n$, and genes $g=1$ to $G$. Write $R_i$ for the total number of mapped reads for sample $i$:
$$R_i=\sum_{g=1}^{G}r_{gi}$$ They define the log-counts per million (log-cpm) value for each count as: $$y_{gi}=\log_2\left(\frac{r_{gi}+0.5}{R_i+1}\times 10^6\right)$$
Write $\lambda=E(r)$ for the expected value of a read count given the experimental conditions, and suppose that: $$\text{Var}(r)=\lambda+\phi\lambda^2$$ If $r$ is large, then the log-cpm value of the observation is: $$y\approx\log_2(r)-\log_2(R)+6\log_2(10)$$ where $R$ is the library size. The analysis is conditional on $R$, so $R$ is treated as a constant. It follows that $\text{Var}(y)\approx \text{Var}(\log_2 r)$. If $\lambda$ also is large, then:
$$(\log_2 r)(\ln 2)\approx \ln r \approx \ln\lambda+\frac{r-\lambda}{\lambda}$$ so $$\text{Var}(y)(\ln 2)^2\approx\frac{\text{Var}(r)}{\lambda^2}=\frac{1}{\lambda}+\phi$$
How should I deduce the last 2 equations?

Based on Taylor expansions for the moments of functions of random variables
$$
\begin{align}
\log X & = \log (\mu_X+X-\mu_X) \\
&\approx \log (\mu_X)+ \log' (\mu_X)(X-\mu_X)+\frac{1}{2}\log''(\mu_X)(X-\mu_X)^2 \\
&\approx\log (E(X))+\frac{X-E(X)}{E(X)}
\end{align}
$$ so take expectations and variances on both sides,
$$E(\log X)\approx\log (E(X))$$
$$\text{Var}(\log X)\approx E\left[\frac{X-E(X)}{E(X)}\right]^2=\frac{\text{Var}(X)}{(E(X))^2}$$
So
$$E[\log r]\approx\log (\lambda)+\frac{r-\lambda}{\lambda}
$$
$$\text{Var}[\log r]\approx\frac{\text{Var}(r)}{\lambda^2}
$$
