ANOVA is valid because your response variable although discrete is quantitative. ANOVA assumes that the probability density function of the mean of your samples is t-student; and, the mean of your samples are not necessarily discrete. In my opinion the log transformation is not necessary.

Do not take logarithm of a count data and apply Anova.

Instead you can just apply Anova which assumes your dependent variable is continuous (which is obviuosly wrong!).

Or, try a Poisson regression which suites best.

Use Poisson model in a GLM procedure in R which is robust for such count outcomes. Then check for overdispersion from residuals of the model. A negative binomial model could be used in case of significant overdispersion.

Count data should be analyzed with log linear models, binnomial, multinomial, or perhaps Poisson. ANOVA is completely inappropriate as it is for continuous data not discrete.

If you do use ANOVA the recommended transformation is usually either the square root or the Freeman-Turkey transformation. These are proposed as variance stabilising transformations for Poisson count data. However, as noted by others, this generally isn't recommended now that software for count models such Poisson regression and related techniques such as negative binomial regression (if there is over or ucnderdispersion) are available.

It is also important to look at the data for potential problems such excess zeroes which may require a zero-inflated count model.

Thank you all for your valuable answers!