All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
The Takbeer (saying Allaahu Akbar) after the obligatory prayers starting from the day of ‘Arafah and during the days of Tashreeq [which are the 11th, 12th and 13th of Thul-Hijjah] was narrated on the authority of a group of Companions, like ‘Umar, ‘Ali, Ibn Mas‘ood, Ibn ‘Abbaas and Jaabir ibn ‘Abdullaah . This was reported from them by Al-Haakim in Al-Mustadrak, Ad-Daaraqutni in As-Sunan, Al-Bayhaqi in As-Sunan Al-Kubra as well as Ibn Abi Shaybah and others.
Besides, the scholars of the Salaf (righteous predecessors) as well as the late and contemporary scholars also said this, and we do not know any scholar who denied it. It is for this reason that when Imaam Ahmad was asked: “Which Hadeeth do you rely on that the Takbeer is from the Fajr prayer on the day of ‘Arafah until the last day of Tashreeq?; he answered: “I rely on the consensus of ‘Umar, ‘Ali, Ibn ‘Abbaas and Ibn Mas‘ood .” This is reported by Ibn Qudaamah in his book Al-Mughni.
Shaykh-ul-Islam Ibn Taymiyyah said in Majmoo' Al-Fataawa: “The most correct view about Takbeer, the opinion of the majority of scholars of the Salaf and the jurists from among the Companions and the Imaams (prominent scholars), is that one should utter the Takbeer starting from the Fajr prayer on the day of ‘Arafah until the end of the days of Tashreeq after every obligatory prayer, and it is permissible for every person to utter the Takbeer loudly when going out to the ‘Eed prayer; this is indeed according to the agreement of the four Imaams (i.e. the Imaams of the Four Schools of jurisprudence).”
In regard to the statement of the group whom you referred to in the question that “the Takbeer must be with the wording 'Allaahu Akbar'” only without any addition, then this is also incorrect. This is because the Takbeer which the Companions used to say was with different wordings which have an addition (to the word Allaahu Akbar). For instance, the Takbeer of Ibn Mas‘ood which was narrated from him by At-Tabaraani and others was with the wording: "Allaahu Akbar, Allaahu Akbar, Laa Ilaaha Illa Allaah, wa-Allaahu Akbar, Allaahu Akbar, walillaahi Al-Hamd” (Allaah is great, Allaah is great, and None has the right to be worshipped but Allaah, Allaah is great, Allaah is great, and all praise be to Allaah).
As regard the Takbeer of Ibn ‘Abbaas which was reported from him by Ibn Abi Shaybah in his Al-Musannaf, it was as follows: “Allaahu Akbaru Kabeera, Allaahu Akbaru Kabeera, Allaahu Akbar wa ajall, Allaahu Akbar walillaahi Al-Hamd.” (Allaah is Most Great, Allaah is Most Great, Allaah is Great and Exalted, Allaah is Great and all praise be to Allaah).
However, the wording reported from Ibn Mas‘ood is the wording reported from most of the Companions . Shaykh-ul-Islam Ibn Taymiyyah said: “The formula of Takbeer that was reported from most of the Companions was attributed to the Prophet, sallallaahu ‘alayhi wa sallam, which is as follows: ''Allaahu Akbar, Allaahu Akbar, Laa Ilaaha Illa Allaah, wa-Allaahu Akbar, Allaahu Akbar, walillaahi Al-Hamd.” [End of quote]
Also, we do not know the evidence of the group whom you mentioned in the question who say that the Takbeer must not be said loudly; we do not know where they got this from as no one before them said as such.
Finally, we remind those people to fear Allaah and we remind them also of the advice of Imaam Ahmad to one of his students when he said to him: “Beware of speaking about any issue on which you do not have an Imaam.”; meaning, ‘do not say any statement which the scholars did not precede you in saying it’.”
Allaah Knows best.