SELAMAT HARI RAYA IDUL FITRI 1446H
\[
\text{SELA. } \log_3\left(\sqrt{2^{{\text{MAT}}+\text{MAT}}}\right) + \frac{ \log_{\sqrt{7}}(16^{\text{HARI}})}{\log_{\sqrt[4]{49} }(81^{\frac{1}{\text{RAYA}}})} = \log_7 \left(\sqrt[3]{25}\right) \cdot \log_3 \left(\sqrt{7^{\text{ID}}}\right) \cdot \log_5(8^{\text{UL}}) + \text{RI. } \log_{\sqrt{27}}\left(\sqrt{8^{\text{FIT}}}\right) + \log_3 (2) \log_e(e^{1446H})
\]
\[\text{SELA. } \log_3\left(\sqrt{2^{\text{MAT} + \text{MAT}}}\right) + \frac{ \log_{\sqrt{7}}(16^{\text{HARI}})}{\log_{\sqrt[4]{49} }(81^{\frac{1}{\text{RAYA}}})}
\]
\[= \log_7 \left(\sqrt[3]{25}\right) \cdot \log_3 \left(\sqrt{7^{\text{ID}}}\right) \cdot \log_5(8^{\text{UL}}) + \text{RI. } \log_{\sqrt{27}}\left(\sqrt{8^{\text{FIT}}}\right) + \log_3 (2) \log_e(e^{1446H})
\]
\[\text{Langkah 1: Sederhanakan Bentuk Logaritma}\]
Bagian Kiri Persamaan
1.
\[\log_3\left(\sqrt{2^{2\text{MAT}}}\right) = \log_3(2^{\text{MAT}}) = \text{MAT} \cdot \log_3 2\]
2.
\[\frac{ \log_{\sqrt{7}}(16^{\text{HARI}})}{\log_{\sqrt[4]{49} }(81^{\frac{1}{\text{RAYA}}})} \]
Pembilang:
\[\log_{\sqrt{7}}(16^{\text{HARI}}) = 8\text{HARI} \frac{\log_3 2}{\log_3 7}\]
Penyebut:
\[\log_{\sqrt[4]{49}}(81^{\frac{1}{\text{RAYA}}}) = \frac{8}{\text{RAYA}} \frac{1}{\log_3 7}\]
Gabungan:
\[\frac{8\text{HARI} \frac{\log_3 2}{\log_3 7}}{\frac{8}{\text{RAYA}} \frac{1}{\log_3 7}} = \text{HARI} \cdot \text{RAYA} \cdot \log_3 2\]
Bagian Kanan Persamaan
3.
\[ \log_7 \left(\sqrt[3]{25}\right) \cdot \log_3 \left(\sqrt{7^{\text{ID}}}\right) \cdot \log_5(8^{\text{UL}}) \]
Sederhanakan:
\[\frac{2}{3} \log_7 5 \cdot \frac{\text{ID}}{2} \log_3 7 \cdot \text{UL} \log_5 8 = \text{ID} \cdot \text{UL} \cdot \log_3 2\]
4.
\[ \text{RI. } \log_{\sqrt{27}}\left(\sqrt{8^{\text{FIT}}}\right) + \log_3 (2) \log_e(e^{1446H}) \]
\[ \log_{\sqrt{27}}\left(\sqrt{8^{\text{FIT}}}\right) \text{RI. } + \log_3 (2) \log_e(e^{1446H}) \]
Sederhanakan:
\[\text{FIT RI} \log_3 2 + 1446H \log_3 2\]
Kesimpulan
\[ (\text{SELA} \text{MAT} + \text{HARI} \cdot \text{RAYA}) \log_3 2 = (\text{ID} \cdot \text{UL} + \text{FIT RI} + 1446H) \log_3 2\]