Which Prayers Are Prayed Loudly?
Question

What is the evidence from the Quran and Sunnah for reciting quietly in Dhuhr and `Asr, when the recitation in Fajr, Maghrib and `Isha’ is done out loud?
Praise be to Allah.Which prayers are prayed loudly?
We thank you for this concern and appreciate your interest in finding out the evidence from the Quran and Sunnah at such a young age, and we ask Allah to make you of benefit to others.
Allah, may He be glorified and exalted, has commanded us to follow the Prophet (blessings and peace of Allah be upon him) and emulate him, as He, may He be exalted, says (interpretation of the meaning):
“There has certainly been for you in the Messenger of Allah an excellent pattern for anyone whose hope is in Allah and the Last Day and [who] remembers Allah often.” [Al-Ahzab 33:21]
The Prophet (blessings and peace of Allah be upon him) said: “Pray as you have seen me praying .” The Prophet (blessings and peace of Allah be upon him) used to recite out loud in Fajr prayer and the first two Rak`ahs of Maghrib and `Isha’, and he would recite quietly in all other cases. 
Evidence for reciting out loud and quietly in the five daily prayers
Texts which refer to reciting out loud include the following:

The report narrated by Al-Bukhari (735) and Muslim (463) from Jubayr ibn Mut`im (may Allah be pleased with him), who said: I heard the Messenger of Allah (blessings and peace of Allah be upon him) reciting [Surat] At-Tur in Maghrib.
The report narrated by Al-Bukhari (733) and Muslim (464) from Al-Bara` (may Allah be pleased with him) who said: I heard the Prophet (blessings and peace of Allah be upon him) reciting “Wat-tini waz-zaytun” [i.e., Surat At-Tin] in `Isha’ prayer, and I never heard anyone with a more beautiful voice than him. 
The report narrated by Al-Bukhari (739) and Muslim (449) from Ibn `Abbas (may Allah be pleased with him) about the jinn coming and listening to the Quran from the Prophet (blessings and peace of Allah be upon him), in which it says that he was leading his companions in Fajr prayer, and when they heard the Quran they listened attentively to it.

These Hadiths indicate that the Prophet (blessings and peace of Allah be upon him) used to recite out loud in such a way that those who were present could hear it.
Evidence that the recitation is to be done quietly in Dhuhr and `Asr prayers includes the following:

The report narrated by Al-Bukhari (713) from Khabbab (may Allah be pleased with him), according to which someone asked him: Did the Messenger of Allah (blessings and peace of Allah be upon him) recite Quran in Dhuhr and `Asr? He said: Yes. We said: How did you know that? He said: From the movement of his beard [as he recited quietly].

Thus it becomes clear that reciting out loud in the prayers where that is to be done and reciting quietly in the prayers where that is to be done is the Sunnah of the Prophet (blessings and peace of Allah be upon him), and the Muslims unanimously agreed on these rulings.
Al-Bukhari (738) and Muslim (396) narrated that Abu Hurayrah (may Allah be pleased with him) said: In every prayer you should recite. What the Messenger of Allah (blessings and peace of Allah be upon him) made us hear we make you hear, and what he recited quietly we recite quietly.
An-Nawawi (may Allah have mercy on him) said:
“The Sunnah is to recite out loud in the (first) two Rak`ahs of Fajr, Maghrib and `Isha’, and in Jumu`ah, and to recite quietly in Dhuhr and `Asr, and the third Rak`ah of Maghrib, and the third and fourth rak`ahs of `Isha’. All of this is according to the consensus of the Muslims, in addition to the authentic Hadiths that all indicate that.” (Al-Majmu` Sharh Al-Muhadhdhab, 3/389)
Ibn Qudamah (may Allah have mercy on him) said:
“Recitation should be done quietly in Dhuhr and `Asr, and out loud in the first two Rak`ahs of Maghrib and `Isha’, and in all of Fajr prayer. … The basis for that is the action of the Prophet (blessings and peace of Allah be upon him), as is proven by the narration of later generations from earlier generations. If a person recites out loud in a place where the recitation should be done quietly, or recites quietly in a place where recitation should be done out loud, he has neglected the Sunnah, although his prayer is valid.” (Al-Mughni, 2/270)
For more details, please see the following answers: 67618 , 26268 , 197414 , 91325 , and  114369 . 
And Allah knows best.