Mehlala ea Lipotso tse Buisanang ka Li-Exponents le Li-Logarithms
Li-exponents le li-logarithm ke likhopolo tse peli tsa bohlokoa tsa lipalo tse atisang ho kopana mafapheng a fapaneng a thuto, joalo ka lipalo, saense, moruo le boenjiniere. Kutloisiso e ntle ea li-exponents le li-logarithm e bohlokoa bakeng sa ho rarolla mathata a fapaneng a lipalo. Sengoloa sena se tla fana ka mehlala ea mathata le lipuisano tse qaqileng tse amanang le li-exponents le li-logarithm.
Sehlahisoa
Sehlomathiso ke nomoro e bontshang hore na nomoro ya motheo e atiswa ka makgetlo a makae ka boyona. Sebopeho se akaretsang sa sehlomathiso ke \(a^n\), moo \(a\) e leng nomoro ya cardinal mme \(n\) e le sehlomathiso.
Mohlala oa Mathata a Hlahisang
Potso ea 1:
Fumana boleng ba \(2^5\).
Puisano:
Boleng ba \(2^5\) ke 2 bo atisitsweng ka boyona ka makgetlo a 5.
\[ 2^5 = 2 \makgetlo a 2 \makgetlo a 2 \makgetlo a 2 \makgetlo a 2 = 32 \]
Kahoo, boleng ba \(2^5\) ke 32.
Potso ea 2:
Bala boleng ba \( (3^2) \makgetlo (3^3) \).
Puisano:
Ho rarolla bothata bona, re ka sebelisa e 'ngoe ea melao ea motheo ea li-exponents e reng:
\[ a^m \times a^n = a^{m+n} \]
E le hore,
\[ (3^2) \makgetlo (3^3) = 3^{2+3} = 3^5 = 243 \]
Kahoo, boleng ba \( (3^2) \times (3^3) \) ke 243.
Potso ea 3:
Nolofatsa \( \frac{5^6}{5^3} \).
Puisano:
Ho nolofatsa dikarolwana tsa exponential tse nang le motheo o tshwanang, re ka sebedisa molao ona:
\[ \frac{a^m}{a^n} = a^{mn} \]
E le hore,
\[ \frac{5^6}{5^3} = 5^{6-3} = 5^3 = 125 \]
Kahoo, boleng ba \( \frac{5^6}{5^3} \) ke 125.
Logarithm
Logarithm ke phetoho ea exponent. Ka kakaretso, haeba \( a^b = c \), joale \( \log_a c = b \). Ka mantsoe a mang, logarithm ea nomoro ke exponent e hlokahalang ho fumana nomoro eo ho tsoa setsing.
Lipotso tsa Mohlala oa Logarithm
Potso ea 4:
Fumana boleng ba \( \log_2 32 \).
Puisano:
Ho fumana boleng ba \( \log_2 32 \), re hloka ho fumana boleng ba exponent e hlahisang 32 ha motheo e le 2.
\[ 2^5 = 32 \]
Ho bolela,
\[ \log_2 32 = 5 \]
Kahoo, boleng ba \( \log_2 32 \) ke 5.
Potso ea 5:
Bala boleng ba \( \log_3 81 \).
Puisano:
Ho fumana boleng ba \( \log_3 81 \), re hloka ho fumana boleng ba exponent e hlahisang 81 ha motheo e le 3.
\[ 3^4 = 81 \]
Ho bolela,
\[ \log_3 81 = 4 \]
Kahoo, boleng ba \( \log_3 81 \) ke 4.
Potso ea 6:
Nolofatsa polelo ea logarithmic \( \log(100) + \log(10) \).
Puisano:
Re ka sebelisa molao oa logarithmic o reng:
\[ \log(a) + \log(b) = \log(ab) \]
E le hore,
\[ \log(100) + \log(10) = \log(100 \makgetlo a 10) = \log(1000) \]
Rea tseba hore 1000 e ka ngoloa e le \( 10^3 \), kahoo:
\[ \log(1000) = \log(10^3) \]
Ho sebelisa melao ea li-logarithm:
\[ \log(10^3) = 3 \]
Kahoo, boleng ba \( \log(100) + \log(10) \) ke 3.
Motsoako oa Li-Exponents le Li-Logarithms
Ka linako tse ling, mathata a lipalo a re hloka ho kopanya tšebeliso ea li-exponents le li-logarithm ho a rarolla.
Lipotso tsa Mehlala ea Motsoako
Potso ea 7:
Haeba \( 2^x = 8 \), fumana boleng ba x.
Puisano:
Ho fumana boleng ba x, re ka ngola 8 ka mokhoa oa exponential ka motheo oa 2.
\[ 8 = 2^3 \]
Kahoo equation e fetoha:
\[ 2^x = 2^3 \]
Kaha metheo e tšoana, li-exponents le tsona li lokela ho tšoana.
\[x = 3 \]
Kahoo, boleng ba x ke 3.
Potso ea 8:
Fumana boleng ba \( \log_5 25 \).
Puisano:
Ho fumana boleng ba \( \log_5 25 \), re hloka ho fumana boleng ba exponent e hlahisang 25 ha motheo e le 5.
\[ 5^2 = 25 \]
Ho bolela,
\[ \log_5 25 = 2 \]
Kahoo, boleng ba \( \log_5 25 \) ke 2.
Potso ea 9:
Haeba \( \log_2 (x^2) = 6 \), fumana boleng ba x.
Puisano:
Ho fumana boleng ba x, re ka fetola equation ea logarithmic hore e be sebopeho sa exponential.
\[ \log_2 ( x^2 ) = 6 \]
ho bolela,
\[ x^2 = 2^6 \]
\[x^2 = 64 \]
Kahoo, re hloka ho fumana boleng ba x bo kgotsofatsang \( x^2 = 64 \).
\[ x = \sqrt{64} \]
\[x = 8 \]
kapa
\[ x = -8 \]
Kahoo, boleng ba x ke 8 kapa -8.
Qetello
Li-exponents le li-logarithm ke likhopolo tsa bohlokoa lipalo. Ka kutloisiso le mokhoa o nepahetseng, re ka rarolla mathata a fapaneng habonolo a kenyeletsang li-exponents le li-logarithm. Mehlala e kaholimo e lebelletsoe ho re thusa ho utloisisa likhopolo tsa motheo tsa li-exponents le li-logarithm le mokhoa oa ho li sebelisa ho rarolla mathata. Ka ho ikoetlisa khafetsa, re tla tloaelana le ho ba le boiphihlelo haholoanyane ho rarolla mathata a lipalo a kenyeletsang li-exponents le li-logarithm.
