{"id":308465,"date":"2022-11-20T17:23:00","date_gmt":"2022-11-20T10:23:00","guid":{"rendered":"https:\/\/quipperhome.wpcomstaging.com\/?p=308465"},"modified":"2023-02-06T10:23:12","modified_gmt":"2023-02-06T03:23:12","slug":"eksponen","status":"publish","type":"post","link":"https:\/\/quipperhome.wpcomstaging.com\/mapel\/matematika\/eksponen\/","title":{"rendered":"Eksponen: Pengertian, Fungsi Contoh, bentuk, rumus"},"content":{"rendered":"\n
\"\"<\/figure>\n\n\n\n

Hai Quipperian, apakah kamu sudah belajar tentang notasi ilmiah? Salah satu ciri khas notasi ilmiah adalah adanya bilangan berpangkat untuk meringkas penulisan sesuatu yang berukuran terlalu besar atau terlalu kecil, misalnya massa elektron 9,1 x 10-31<\/sup> kg. Ternyata, bilangan berpangkat semacam itu juga dipelajari di Matematika, lho<\/em>. Namun, kamu akan mengenalnya sebagai eksponen. Apa yang dimaksud dengan eksponen? Daripada penasaran, yuk simak selengkapnya!<\/p>\n\n\n\n

Pengertian Eksponen<\/strong><\/h2>\n\n\n\n

Eksponen adalah bentuk perkalian suatu bilangan dengan dirinya sendiri secara berulang-ulang. Adapun bentuk umum eksponen atau rumus eksponen adalah sebagai berikut.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

a<\/em>b<\/sup><\/em>, dengan syarat a<\/em> \u2260 1 dan b<\/em> \u03f5 R <\/p>\n\n\n\n

Dari penulisan bentuk di atas, a<\/em> disebut sebagai basis atau bilangan pokok dasar, sedangkan b<\/em> disebut sebagai pangkat atau eksponen. Jika b<\/em> termasuk bilangan bulat positif, maka a<\/em>b<\/sup><\/em> bisa dinyatakan seperti berikut.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Sifat-Sifat Eksponen<\/strong><\/h2>\n\n\n\n

Operasi bentuk perpangkatan atau eksponen tentu berbeda dengan bilangan biasa. Namun, kamu tak perlu khawatir karena operasi itu mengacu pada sifat-sifat eksponen berikut ini.<\/p>\n\n\n\n

Sifat penjumlahan pangkat<\/strong><\/h3>\n\n\n\n

Sifat penjumlahan pangkat hanya berlaku jika kamu mengalikan dua eksponen atau lebih dengan basis yang sama. Lantas, bagaimana jika basisnya tidak sama? Tentu tidak berlaku sifat ini, ya. Perhatikan contoh berikut.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Sifat pengurangan pangkat<\/strong><\/h3>\n\n\n\n

Sifat pengurangan pangkat hanya berlaku jika kamu melakukan pembagian antara dua eksponen atau lebih dengan basis yang sama. Sama seperti sifat penjumlahan pangkat, sifat ini hanya berlaku untuk basis yang sama. Perhatikan contoh berikut.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Sifat perkalian pangkat<\/strong><\/h3>\n\n\n\n

Sifat perkalian pangkat berlaku jika suatu eksponen dipangkatkan lagi. Perhatikan contoh berikut.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Sifat pembagian pangkat<\/strong><\/h3>\n\n\n\n

Sifat pembagian pangkat berlaku jika suatu eksponen berada di dalam bentuk akar atau kamu bisa menyebutnya bentuk akar eksponen. Apakah kamu masih ingat penulisan bentuk akar menjadi eksponen? Perhatikan contoh berikut.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Sifat pangkat nol<\/strong><\/h3>\n\n\n\n

Jika kamu menjumpai bilangan yang dipangkatkan nol, kamu bisa langsung menuliskannya sebagai 1. Mengapa demikian? Karena bilangan yang dipangkatkan nol, berapapun basisnya akan sama dengan satu (a<\/em> \u2260 0). Perhatikan contoh berikut.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Jadi, kamu tidak perlu repot-repot menghitungnya ya.<\/p>\n\n\n\n

Sifat pangkat satu<\/strong><\/h3>\n\n\n\n

Jika suatu bilangan dipangkatkan satu hasilnya sama dengan bilangan itu sendiri. Agar tidak salah, kamu harus jeli membedakannya dengan sifat pangkat nol. Perhatikan contoh berikut.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Sifat pangkat negatif<\/strong><\/h3>\n\n\n\n

Sifat pangkat negatif artinya satu per perkalian berulang suatu bilangan dengan dirinya sendiri. Sifat ini kebalikan dari perpangkatan positif. Perhatikan contoh berikut. <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Dari contoh di atas, dapat diketahui bahwa perpangkatan negatif akan menghasilkan suatu pecahan.<\/p>\n\n\n\n

Fungsi Eksponen<\/strong><\/h2>\n\n\n\n

Salah satu penerapan bilangan berpangkat di dalam Matematika adalah digunakan pada fungsi eksponen. Di artikel sebelumnya, Quipper Blog sudah pernah membahas tentang persamaan eksponen. Saat kamu belajar persamaan eksponen, pasti akan bertemu fungsi eksponen di dalamnya. Lalu, apa itu fungsi eksponen? Fungsi eksponen adalah suatu fungsi yang memuat variabel di bagian pangkatnya.  Adapun bentuk umum fungsi eksponen adalah sebagai berikut.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Jika kamu biasa mengenal fungsi bervariabel x<\/em>, maka persamaan di atas juga bisa diubah dalam variabel x<\/em>, sehingga menjadi:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Perhatikan contoh berikut.<\/p>\n\n\n\n

Suatu fungsi eksponen dinyatakan sebagai f<\/em>(x<\/em>) = 23<\/sup>x<\/sup><\/em>+1<\/sup>. Tentukan nilai fungsi tersebut untuk x<\/em> = 1.<\/p>\n\n\n\n

Pembahasan:<\/p>\n\n\n\n

Untuk menyelesaikan soal tersebut, kamu harus mensubstitusikan nilai x<\/em> = 1 pada variabel x<\/em> di bagian fungsinya.<\/p>\n\n\n\n

f<\/em>(x<\/em>) = 23<\/sup>x<\/sup><\/em>+1<\/sup><\/p>\n\n\n\n

\u2194 f<\/em>(1) = 23(1)+1<\/sup><\/p>\n\n\n\n

\u2194f<\/em>(1) = 24<\/sup> = 16.<\/p>\n\n\n\n

Grafik Fungsi Eksponen<\/strong><\/h2>\n\n\n\n

Grafik fungsi eksponen diperoleh melalui plot titik f<\/em>(x<\/em>) untuk setiap himpunan nilai x <\/em>pada koordinat Cartesius. Grafik ini cukup mudah dikenali, yaitu berupa kurva lengkung yang permukaannya mulus. Secara umum, grafik fungsi eksponen dibagi menjadi empat pola berdasarkan Perhatikan contoh berikut.<\/p>\n\n\n\n

Buatlah grafik fungsi f<\/em>(x<\/em>) = 2x<\/sup><\/em>!<\/p>\n\n\n\n

Pembahasan:<\/p>\n\n\n\n

Mula-mula, kamu harus menentukan rentang nilai x<\/em> dan f<\/em>(x<\/em>) yang memenuhi dalam bentuk tabel seperti berikut.<\/p>\n\n\n\n

x<\/em><\/th>f<\/em>(x<\/em>)<\/th><\/tr><\/thead>
-2<\/td><\/td><\/tr>
-1<\/td><\/td><\/tr>
0<\/td>1<\/td><\/tr>
1<\/td>2<\/td><\/tr>
2<\/td>3<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Jika diplot ke dalam koordinat Cartesius, data di atas akan menjadi seperti berikut.<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Dari grafik di atas, terlihat bahwa jika fungsi eksponennya berpangkat positif, semakin besar nilai x<\/em>, semakin besar pula f<\/em>(x<\/em>). Hal itu ditandai dengan semakin ke kanan, kurva akan semakin naik. Grafik seperti ini disebut sebagai grafik monoton naik. Lantas, bagaimana jika fungsi eksponennya berpangkat negatif? Yuk, kita buat grafiknya.<\/p>\n\n\n\n

Misal kamu ambil f<\/em>(x<\/em>) = 2-x<\/sup><\/em>!<\/p>\n\n\n\n

Mula-mula, kamu buat tabel seperti contoh sebelumnya.<\/p>\n\n\n\n

x<\/em><\/th>f<\/em>(x<\/em>)<\/th><\/tr><\/thead>
-2<\/td>4<\/td><\/tr>
-1<\/td>2<\/td><\/tr>
0<\/td>1<\/td><\/tr>
1<\/td><\/td><\/tr>
2<\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Jika diplot dalam bentuk grafik, menjadi seperti berikut.<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Dari grafik di atas, semakin besar nilai x<\/em>, semakin kecil nilai f<\/em>(x<\/em>). Hal itu ditandai dengan semakin ke kanan, kurva semakin landai mendekati nol. Itulah mengapa, grafik ini disebut sebagai grafik monoton turun. Adapun persamaan dari kedua grafik adalah keduanya sama-sama memotong sumbu-y di koordinat (0, 1).<\/p>\n\n\n\n

Ternyata, bentuk grafik seperti itu berlaku jika basisnya berupa bilangan positif, lho<\/em>. Lalu, bagaimana jika basisnya negatif? Silahkan kamu coba di rumah, ya.<\/p>\n\n\n\n

Contoh Soal Eksponen<\/strong><\/h2>\n\n\n\n

Untuk mengasah pemahamanmu tentang materi ini, yuk simak contoh soal berikut.<\/p>\n\n\n\n

Contoh Soal 1<\/strong><\/h3>\n\n\n\n

Diketahui 42020<\/sup> \u2013 3. 42019<\/sup> = a<\/em>b<\/sup>.<\/p>\n\n\n\n

Berapakah nilai 2a<\/em> + b?<\/p>\n\n\n\n

Pembahasan:<\/p>\n\n\n\n

Mula-mula, kamu harus menguraikan bentuk perpangkatan di atas.<\/p>\n\n\n\n

42020<\/sup> \u2013 3. 42019<\/sup> = a<\/em>b<\/sup><\/p>\n\n\n\n

\u2194 4 \u00d7 42019<\/sup> \u2013 3 \u00d7 42019<\/sup> = a<\/em>b<\/sup><\/p>\n\n\n\n

Selanjutnya, kamu bisa memisalkan 42019<\/sup> sebagai x<\/em>, sehingga:<\/p>\n\n\n\n

4 \u00d7 42019<\/sup> \u2013 3 \u00d7 42019<\/sup> = a<\/em>b<\/sup><\/p>\n\n\n\n

\u21944x<\/em> \u2013 3x<\/em> = a<\/em>b<\/sup><\/p>\n\n\n\n

\u2194 x<\/em> = a<\/em>b<\/sup><\/p>\n\n\n\n

\u2194 42019<\/sup> = a<\/em>b<\/sup><\/em><\/p>\n\n\n\n

Dengan demikian, a<\/em> = 4 dan b<\/em> = 2019.<\/p>\n\n\n\n

Jadi, nilai 2a<\/em> + b<\/em> = 2(4) + 2019 = 2027.<\/p>\n\n\n\n

Contoh Soal 2<\/strong><\/h3>\n\n\n\n

Jika f<\/em>(x<\/em>) = 32<\/sup>x<\/sup><\/em> + 4<\/sup>, maka berapakah nilai f<\/em>(-2) + 2f<\/em>(0)?<\/p>\n\n\n\n

Pembahasan:<\/p>\n\n\n\n

Mula-mula, kamu harus mencari nilai fungsi untuk x<\/em> = -2 dan x<\/em> = 0.<\/p>\n\n\n\n

f<\/em>(-2) = 32(-2) + 4<\/sup><\/p>\n\n\n\n

       = 3-4 + 4<\/sup><\/p>\n\n\n\n

       = 30<\/sup> = 1<\/p>\n\n\n\n

f<\/em>(0) = 32(0) + 4<\/sup><\/p>\n\n\n\n

       = 30 + 4<\/sup><\/p>\n\n\n\n

       = 34<\/sup> = 81<\/p>\n\n\n\n

Jadi, nilai f<\/em>(-2) + 2f<\/em>(0) = 1 + 2(81) = 163.<\/p>\n\n\n\n

Contoh Soal 3<\/strong><\/h3>\n\n\n\n

Bagaimanakah bentuk grafik fungsi f<\/em>(x<\/em>) = 5–<\/sup>x<\/sup><\/em> +1?<\/p>\n\n\n\n

Pembahasan:<\/p>\n\n\n\n

Mula-mula, tentukan interval untuk nilai menggunakan SUPER \u201cSolusi Quipper\u201d berikut ini.<\/p>\n\n\n\n

x<\/em><\/strong><\/th>f<\/em>(x<\/em>)<\/strong><\/th>Koordinat<\/strong><\/th><\/tr><\/thead>
-1<\/td>6<\/td>(-1,6)<\/td><\/tr>
0<\/td>2<\/td>(0,2)<\/td><\/tr>
1<\/td>1,2<\/td>(1;1,3)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
x<\/em><\/strong><\/td>f<\/em><\/strong>(<\/strong>x<\/em><\/strong>)<\/strong><\/td>Koordinat<\/strong><\/td><\/tr>
-1<\/td>6<\/td>(-1. 6)<\/td><\/tr>
0<\/td>2<\/td>(0, 2)<\/td><\/tr>
1<\/td>1,2<\/td>(1; 1,2)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Lalu, plot koordinat di atas dalam bentuk grafik seperti berikut.<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n

Sampai sini, apakah Quipperian sudah paham?<\/p>\n\n\n\n

Itulah pembahasan Quipper Blog tentang eksponen. Semoga bermanfaat, ya. Jika kamu ingin mendapatkan materi lengkapnya, silahkan gabung Quipper Video<\/a>. Salam Quipper!<\/p>\n","protected":false},"excerpt":{"rendered":"

Hai Quipperian, apakah kamu sudah belajar tentang notasi ilmiah? Salah satu ciri khas notasi ilmiah adalah adanya bilangan berpangkat untuk meringkas penulisan sesuatu yang…<\/p>\n","protected":false},"author":156447303,"featured_media":308472,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[679384865],"tags":[679386732],"ppma_author":[679386823,679386836],"class_list":["post-308465","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-matematika","tag-matematika-peminatan-kelas-10"],"acf":[],"yoast_head":"\nEksponen: Pengertian, Fungsi Contoh, bentuk, rumus - Quipper Blog<\/title>\n<meta name=\"description\" content=\"Eksponen adalah bentuk perkalian suatu bilangan dengan dirinya sendiri secara berulang-ulang. 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