Nā nīnau hoʻohālike e kūkākūkā ana i nā hana Logarithmic
He manaʻo koʻikoʻi nā Logarithms i ka makemakika, ʻoi aku hoʻi i ka algebra a me ka nānā ʻana. Pili loa lākou i nā exponents a hoʻohana pinepine ʻia e hoʻoponopono i nā hoʻohālikelike exponential a ma nā noi ʻepekema a me nā ʻenekinia like ʻole. E kūkākūkā kēia ʻatikala i kekahi mau pilikia logarithm i loaʻa pinepine ʻia, me kahi wehewehe piha o kēlā me kēia pilikia.
Hoʻolauna i nā Logarithms
ʻO nā Logarithms ka hoʻohuli o nā exponents. Inā loaʻa iā mākou ka hoohalike exponential \(b^y = x\), a laila ʻo kona ʻano logarithmic ʻo \(y = \log_b{x}\), ʻo ia hoʻi "ʻo y ka logarithm o x me ke kumu b". ʻO kekahi mau logarithms i hoʻohana pinepine ʻia ʻo ia ka logarithm kūlohelohe (kumu \(e\)) a me ka logarithm decimal (kumu 10).
Nā Waiwai o nā Logarithms
Eia kekahi mau waiwai kumu o nā logarithms i hoʻohana pinepine ʻia i ka hoʻoponopono ʻana i nā pilikia:
1. Logarithm o ka huahana:
\[
\log_b{(xy)} = \log_b{x} + \log_b{y}
\]
2. Logarithm o ka quotient:
\[
\log_b{(\frac{x}{y})} = \log_b{x} – \log_b{y}
\]
3. Logarithm o ka exponent:
\[
\log_b{(x^a)} = a \cdot \log_b{x}
\]
4. Hoʻololi o ke kumu logarithmic:
\[
\log_b{x} = \frac{\log_k{x}}{\log_k{b}}
\]
Nā Nīnau Laʻana a me ke Kūkākūkā
1. Nīnau 1:
E huli i ka waiwai o \( \log_2{32} \).
Kūkākūkā:
Ua ʻike mākou hiki ke kākau ʻia ʻo \(32\) ma ke ʻano he \(2^5\). No laila:
\[
\log_2{32} = \log_2{(2^5)} = 5 \cdot \log_2{2}
\]
ʻOiai ʻo \(\log_2{2} = 1\):
\[
\log_2{32} = 5 \cdot 1 = 5
\]
No laila, ʻo ka waiwai o \( \log_2{32} \) he 5.
2. Nīnau 2:
Inā ʻo \( \log_3{x} = 4 \), e ʻimi i ka waiwai o \( x \).
Kūkākūkā:
Ma muli o ka wehewehe ʻana o ka logarithm, hiki ke kākau hou ʻia ʻo \( \log_3{x} = 4 \) ma ke ʻano exponential:
\[
3^4 = x
\]
Ke helu nei i ka \(3^4\):
\[
3 ^ 4 = 81
\]
No laila, ʻo ka waiwai o \( x \) he 81.
3. Nīnau 3:
Ua hāʻawi ʻia kahi hoohalike \( \log_{10}{x} = -2 \). E huli i ka waiwai o \( x \).
Kūkākūkā:
E hoʻololi i ke ʻano logarithmic i ke ʻano exponential:
\[
10^{-2} = x
\]
Ke helu nei i ka \(10^{-2}\):
\[
10^{-2} = \frac{1}{10^2} = \frac{1}{100} = 0.01
\]
No laila, ʻo ka waiwai o \( x \) he 0.01.
4. Nīnau 4:
E huli i ka waiwai o \( \log_5{(125 \cdot 25)} \).
Kūkākūkā:
Ua ʻike mākou ʻo \(125 = 5^3\) a me \(25 = 5^2\). A laila:
\[
\log_5{(125 \cdot 25)} = \log_5{(5^3 \cdot 5^2)}
\]
Ma muli o nā waiwai o ka huahana o nā logarithms:
\[
\log_5{(5^3 \cdot 5^2)} = \log_5{5^5}
\]
Ke hoʻohana nei i nā waiwai o nā mana logarithmic:
\[
\log_5{5^5} = 5 \cdot \log_5{5}
\]
ʻOiai ʻo \(\log_5{5} = 1\):
\[
5 \cdot 1 = 5
\]
No laila, ʻo ka waiwai o \( \log_5{(125 \cdot 25)} \) he 5.
5. Nīnau 5:
E huli i ka waiwai o \( \log_{2}{(8 \cdot \sqrt{2})} \).
Kūkākūkā:
Ua ʻike mākou ʻo \(8 = 2^3\) a me \(\sqrt{2} = 2^{1/2}\). A laila:
\[
\log_{2}{(8 \cdot \sqrt{2})} = \log_{2}{(2^3 \cdot 2^{1/2})}
\]
Ma muli o nā waiwai o ka huahana o nā logarithms:
\[
\log_{2}{(2^3 \cdot 2^{1/2})} = \log_{2}{(2^{3 + 1/2})} = \log_{2}{(2^{3.5})}
\]
Ke hoʻohana nei i nā waiwai o nā mana logarithmic:
\[
\log_{2}{(2^{3.5})} = 3.5 \cdot \log_{2}{2}
\]
ʻOiai \(\log_{2}{2} = 1\):
\[
3.5 \cdot 1 = 3.5
\]
No laila, ʻo ka waiwai o \( \log_{2}{(8 \cdot \sqrt{2})} \) he 3.5.
6. Nīnau 6:
Inā \( \log_4{y} – \log_4{2} = 3 \), e ʻimi i ka waiwai o \( y \).
Kūkākūkā:
Ma muli o nā waiwai o ka logarithmic quotient:
\[
\log_4{(\frac{y}{2})} = 3
\]
E hoʻololi i ke ʻano logarithmic i ka exponential:
\[
4^3 = \frac{y}{2}
\]
Ke helu nei i ka \(4^3\):
\[
4 ^ 3 = 64
\]
No laila:
\[
64 = \frac{y}{2}
\]
No laila:
\[
y = 64 \cdot 2 = 128
\]
No laila, ʻo ka waiwai o \(y \) he 128.
7. Nīnau 7:
E huli i ka waiwai o \( \log_{6}{\frac{1}{36}} \).
Kūkākūkā:
Ua ʻike mākou \(36 = 6^2\). A laila:
\[
\log_{6}{\frac{1}{36}} = \log_{6}{(6^{-2})}
\]
Ke hoʻohana nei i nā waiwai o nā mana logarithmic:
\[
\log_{6}{(6^{-2})} = -2 \cdot \log_{6}{6}
\]
ʻOiai \(\log_{6}{6} = 1\):
\[
-2 \cdot 1 = -2
\]
No laila, ʻo ka waiwai o \( \log_{6}{\frac{1}{36}} \) he -2.
Ka hopena
He mea hana makemakika pono loa nā Logarithms i nā ʻano hana ʻepekema a me nā ʻenekinia like ʻole. ʻO ka hoʻomaopopo ʻana i nā waiwai kumu o nā logarithms hiki ke maʻalahi i ka hoʻoponopono ʻana i nā pilikia he nui. Ua hōʻike kēia ʻatikala i kekahi mau pilikia a ua kūkākūkā i nā logarithms e kū pinepine mai ana ma nā ʻano like ʻole. ʻO ka hoʻomaʻamaʻa ʻana a me ka hoʻomaopopo ʻana i kēia mau manaʻo e kōkua nui i ka hoʻomaopopo ʻana i ke kumuhana o nā logarithms.
