a question of normality/confidence intervals

Hello. Imagine that you have data on wages for a number of individuals, and the distribution is right skewed, which means that it is not normally distributed and confidence intervals can not be used in the normal way.

Imagine now that you take the natural logarithm of all wages and that they now are normally distributed, and from this you form a confidence intervals for the logarithm of wage (which has the point estimate "mean of logWage" +- ConfidenceLevel * standard deviation) which gives you a lower limit and an upper limit.

If you now raise the lower and upper limit by taking e^(lower limit) and e^(upper limit) and end up with a confidence interval for the initial wage variable (and due to skewness, the lower and upper limit has different distances to the point estimate, in contrast to "normal" confidence intervals where the upper and lower limit are of equal distance to the point estimate). Can you consider this to be a confidence interval for the true mean of the wage variable (with all assumptions needed for building a confidence interval)?
If so, would this be sort of a "skewness adjusted confidence interval"?

Thanks.