How to back-transform confidence intervals and standard errors (log2)?

Question

I had skewed normalized count data. So I $log_2$-transformed the data, to get them normally distributed before doing regression — the usual approach. This is the result, which can be considered as the $log_2$-fold-change:

> cf
             Estimate Std. Error   t value    Pr(>|t|)       2.5 %    97.5 %
(Intercept) 0.4120023  0.3752519 1.0979353 0.314327963 -0.50620600 1.3302106
x.1         0.1413944  0.2488691 0.5681476 0.590547229 -0.46756644 0.7503552
x.2         0.6022825  0.2215433 2.7185763 0.034708337  0.06018544 1.1443795
x.3         0.8748745  0.1597857 5.4752979 0.001550213  0.48389290 1.2658561

To back-transform the estimates, in order to present them more intuitive, I can raise 2 to the power of the estimates, to get the fold-change.

> 2^cf[, 1]
(Intercept)         x.1         x.2         x.3 
   1.330531    1.102971    1.518116    1.833849 

But what do I do with the confidence intervals?

$2^{\text{sd}(\log2(Y))} = \text{sd}(Y)$ is probably not the case, but I'm aiming at an approximation.

This looks reasonable;

> cbind(2^cf[, 1] - 2^cf[, 2], 2^cf[, 1] + 2^cf[, 2])
                   [,1]     [,2]
(Intercept)  0.03346515 2.627597
x.1         -0.08530467 2.291246
x.2          0.35213621 2.684097
x.3          0.71672736 2.950970
> cf[, 5:6]
                  2.5 %    97.5 %
(Intercept) -0.50620600 1.3302106
x.1         -0.46756644 0.7503552
x.2          0.06018544 1.1443795
x.3          0.48389290 1.2658561

x.1 still includes zero for instance, but I'm not sure if this is correct.

Maybe confidence intervals need to be re-calculated by transforming the standard errors.

---

Data:

Here my toy data for R, if needed:

set.seed(42)
df <- data.frame(x=matrix(log2(runif(10*3)), 10, 3))
df$y <- as.matrix(cbind(1, df)) %*% runif(4) + rnorm(10,,.5)
fit <- lm(y ~ ., df)
cf <- cbind(coef(summary(fit)), confint(fit))