How to understand that this earth is made up of pancha bootha? Please explain logically. Also, what is the relationship between the five elements akasha, vayu, agni, water, earth?
Your question is how to explain that earth is made up of PanchaMahabhutas . Earth itself is one the Panchamahabhutas . This is very interesting question. So first of all let's understand that there are two earth elements one is subtle and one (on which we live) is gross element.
As suggested by user Rickross , the answer is present in Panchikarana Vartik ,पन्चिकरण वर्तिक or textual explanation by Sri Sureshwaracharya of actual text manual for Paramahansa order by Adi Shankaracharya himself.Sri Sureshwaracharya was direct disciple of Adi Shankaracharya.
According to theory of Panchikarana in the beginning there was Brahman alone , from him came Avyakta from it Mahat-Tattva from it Ahamkara from it Shabda-Pancha-Tanmatra from it subtle PanchaMahaBhuta and from the Subtle panchamahabhutas five gross PanchaMahaBhutas and from them the whole universe came into existence .
Panchikarana involves one half of the original subtle element to be mixed up with 1/8th part each of the other original subtle elements to produce the gross element of the subtle element.
So from the text we come to know that there are two PanchaMahaBhutas and Two earths. One is in subtle form and another in gross form ( which is part of universe).
आसीदेकं परं ब्रह्म नित्यमुक्तविक्रियंम्
तत्स्वमायासमवेशाद् बीजमव्याकृतम ||2 ||
At first , Changeless , ever free , non dual Brahman only was there . By his power of Maya he crafted a Uncultivated seed of this universe.
तस्मादाकाशमुत्पन्नं शब्दतन्मात्ररूपकम ||
स्पर्शात्मकस्ततो वायुस्तेजो रुपत्मकं तत: ||3||
From that Uncultivated seed Akasha (ether) which is of form
Shabda Tanmantra emerged. From that Akasha came the Vayu (air) which is form of Sparsha (touch) . From Vayu there came into existence Teja ( glitter , luster) in the feature of rupa.
आपो रसात्मिकास्तस्मात्ताभ्यो गन्धात्मिका मही |
शब्दैकगुणमाकाशं शब्दस्पर्शगुणो मरुत ||4 ||
From Teja came the water (Jal) , of feature Rasa. From water there came the Prithvi (earth) having feature of smell. Akasha or ether is the only element having one Shabda (sound) feature. Vayu (air ) is having two features Shabda (sound) and Sparsha (touch) .
शब्दस्पर्शरूपगुणं त्रिगुणं तेज उच्चते |
शब्दस्पर्शरुपरसगुणैरापश्चतुर्गुणा: ||5 ||
Tej or luster is having three features and water is having four features Shabda , Sparsha , rupa Rasa and Gandha.
शब्दस्पर्शरूपरसगन्धै: पञ्चगुणा मही |
तेभ्य: समभवत्सुत्रं लिङ्गं सर्वात्मकं महत् ||6 ||
Prithvi is having five forms Shabda
,Sparsha , Rupa , Rasa and Gandha ( Sound , Touch , Form , essence , Scent ) .And from these five microscopic subtle Pancha-Maha-Bhutas ( Shabda ,Sparsha , Rupa , Rasa and Gandha ) came the Mahalinga , sutratma " Hiranyagarbha"
तत: स्थूलानि भूतानि , पञ्च तेभ्यो विराडभूत् |
पञ्चिकृतानि भूतानि , स्थूलानीत्युच्यते बुधै : || 7 ||
From this Sutratma or Hiranyagarbha five gross bhootas came into existence . From these five gross Panchamahabhutas there came Virat.
The theory of Panchikarana is brief is as thus. We have already seen the formation of five subtle PanchaMahaBhutas from Brahman . Panchikarana means dividing each of these 5 subtle bhutas into two , then by keeping one part from each of them aside , again dividing the second remaining part by four. And adjoining each of these parts with other elements except themselves.
So if we look at the verse no. 2 we can find that Earth or Pritivi is firstly also is a subtle element which are forming the base of this universe. And this Prithvi element is in its microscopic form only not as gross or solid.
Verse 6 tells us that from such 5 subtle panchamahabhutas there came the Sutratma or Hiranyagarbha , and from that Hiranyagarbha came the 5 gross Panchamahabhutas.
So basically earth as the gross PanchaMahaBhuta came into existence from combination of these five subtle PanchaMahaBhutas.