Ngā Tauira Pātai e Matapaki ana i ngā Mahi Logarithmic
He ariā matua ngā taupūnga i roto i te pāngarau, inā koa i roto i te arapūnga me te tātari. He hononga tata ēnei ki ngā taupūnga, ā, he maha ngā whakamahinga hei whakaoti rapanga taupūnga me ngā tono pūtaiao me te hangarau. Ka matapakihia e tēnei tuhinga ētahi raruraru taupūnga e kitea pinepinetia ana, me tētahi whakamārama whānui mō ia raruraru.
He Kupu Whakataki ki ngā Pūrākau
Ko ngā tākaritimi te whakahurihanga o ngā taupū. Mēnā kei a tātou te whārite taupū \(b^y = x\), ko tōna āhua tākaritimi ko \(y = \log_b{x}\), ko tōna tikanga "ko y te tākaritimi o x me te pūtake b". Ko ētahi o ngā tākaritimi e whakamahia whānuitia ana ko te tākaritimi maori (pūtake \(e\)) me te tākaritimi ira (pūtake 10).
Ngā Āhuatanga o ngā Logarithms
Ko ngā āhuatanga matua o ngā logarithm e whakamahia ana i roto i te whakaoti rapanga e whai ake nei:
1. Te taupūnga o te hua:
\[
\log_b{(xy)} = \log_b{x} + \log_b{y}
\]
2. Te taupūnga o te haurua:
\[
\log_b{(\frac{x}{y})} = \log_b{x} – \log_b{y}
\]
3. Te taupūnga o te taupūnga:
\[
\log_b{(x^a)} = a \cdot \log_b{x}
\]
4. Te huringa o te pūtake taupū:
\[
\log_b{x} = \frac{\log_k{x}}{\log_k{b}}
\]
Ngā Pātai Tauira me te Kōrero
1. Pātai 1:
Kimihia te uara o \( \log_2{32} \).
Kōrero:
E mōhio ana tātou ka taea te tuhi i te \(32\) hei \(2^5\). Nō reira:
\[
\log_2{32} = \log_2{(2^5)} = 5 \cdot \log_2{2}
\]
Mai i te mea ko \(\log_2{2} = 1\):
\[
\log_2{32} = 5 \cdot 1 = 5
\]
Nō reira, ko te uara o \( \log_2{32} \) he 5.
2. Pātai 2:
Mena ko \( \log_3{x} = 4 \), kimihia te uara o \( x \).
Kōrero:
I runga i te whakamāramatanga o te logarithm, ka taea te tuhi anō i te \( \log_3{x} = 4 \) ki te āhua taupūpū:
\[
3^4 = x
\]
Te tatau i te \(3^4\):
\[
3^4 = 81
\]
Nō reira, ko te uara o \( x \) he 81.
3. Pātai 3:
Ka hoatu he whārite \( \log_{10}{x} = -2 \). Kimihia te uara o \( x \).
Kōrero:
Tahurihia te āhua o te taupūnga ki te āhua o te taupūnga:
\[
10^{-2} = x
\]
Te tatau i te \(10^{-2}\):
\[
10^{-2} = \frac{1}{10^2} = \frac{1}{100} = 0.01
\]
Nō reira, ko te uara o \( x \) he 0.01.
4. Pātai 4:
Kimihia te uara o \( \log_5{(125 \cdot 25)} \).
Kōrero:
E mōhio ana tātou ko \(125 = 5^3\) me \(25 = 5^2\). Kātahi:
\[
\log_5{(125 \cdot 25)} = \log_5{(5^3 \cdot 5^2)}
\]
I runga i ngā āhuatanga o te hua o ngā logarithm:
\[
\log_5{(5^3 \cdot 5^2)} = \log_5{5^5}
\]
Te whakamahi i ngā āhuatanga o ngā mana logarithmic:
\[
\log_5{5^5} = 5 \cdot \log_5{5}
\]
Mai i te mea ko \(\log_5{5} = 1\):
\[
5 \cdot 1 = 5
\]
Nō reira, ko te uara o \( \log_5{(125 \cdot 25)} \) he 5.
5. Pātai 5:
Kimihia te uara o \( \log_{2}{(8 \cdot \sqrt{2})} \).
Kōrero:
E mōhio ana tātou ko \(8 = 2^3\) me \(\sqrt{2} = 2^{1/2}\). Kātahi:
\[
\log_{2}{(8 \cdot \sqrt{2})} = \log_{2}{(2^3 \cdot 2^{1/2})}
\]
I runga i ngā āhuatanga o te hua o ngā logarithm:
\[
\log_{2}{(2^3 \cdot 2^{1/2})} = \log_{2}{(2^{3 + 1/2})} = \log_{2}{(2^{3.5})}
\]
Te whakamahi i ngā āhuatanga o ngā mana logarithmic:
\[
\log_{2}{(2^{3.5})} = 3.5 \cdot \log_{2}{2}
\]
Mai i te mea ko \(\log_{2}{2} = 1\):
\[
3.5 \cdot 1 = 3.5
\]
Nō reira, ko te uara o \( \log_{2}{(8 \cdot \sqrt{2})} \) he 3.5.
6. Pātai 6:
Mena ko \( \log_4{y} – \log_4{2} = 3 \), kimihia te uara o \( y \).
Kōrero:
I runga i ngā āhuatanga o te haurua logarithmic:
\[
\log_4{(\frac{y}{2})} = 3
\]
Tahurihia te āhua o te taupūnga ki te āhua o te taupūnga:
\[
4^3 = \frac{y}{2}
\]
Te tatau i te \(4^3\):
\[
4^3 = 64
\]
Nā reira:
\[
64 = \frac{y}{2}
\]
Nā reira:
\[
y = 64 \cdot 2 = 128
\]
Nō reira, ko te uara o \(y \) he 128.
7. Pātai 7:
Kimihia te uara o \( \log_{6}{\frac{1}{36}} \).
Kōrero:
E mōhio ana tātou ko \(36 = 6^2\). Kātahi:
\[
\log_{6}{\frac{1}{36}} = \log_{6}{(6^{-2})}
\]
Te whakamahi i ngā āhuatanga o ngā mana logarithmic:
\[
\log_{6}{(6^{-2})} = -2 \cdot \log_{6}{6}
\]
Mai i te mea ko \(\log_{6}{6} = 1\):
\[
-2 \cdot 1 = -2
\]
Nō reira, ko te uara o \( \log_{6}{\frac{1}{36}} \) he -2.
Whakamutunga
He taputapu pāngarau tino whai hua ngā logarithm i roto i ngā momo mahi pūtaiao me te hangarau. Mā te mārama ki ngā āhuatanga taketake o ngā logarithm ka māmā ake te whakaoti rapanga. Kua whakarārangihia e tēnei tuhinga ētahi raruraru, ā, kua kōrerohia hoki ngā logarithm e puta pinepine ana i roto i ngā horopaki rerekē. Mā te whakaharatau me te mārama ki ēnei ariā ka tino āwhina i te mōhio ki te kaupapa o ngā logarithm.
