Cov Lus Nug Piv Txwv Sib Tham Txog Cov Exponents thiab Logarithms
Cov exponents thiab logarithms yog ob lub tswv yim lej tseem ceeb uas feem ntau ntsib hauv ntau qhov kev kawm, xws li lej, kev tshawb fawb, kev lag luam, thiab kev tsim kho. Kev nkag siab zoo txog cov exponents thiab logarithms yog qhov tseem ceeb rau kev daws ntau yam teeb meem lej. Tsab xov xwm no yuav muab cov piv txwv teeb meem thiab kev sib tham ntxaws ntxaws txog cov exponents thiab logarithms.
Tus lej qhia
Tus lej exponent yog tus lej uas qhia tias tus lej pib raug muab nws tus kheej ntau npaum li cas. Feem ntau, tus lej exponent yog \(a^n\), qhov twg \(a\) yog tus lej cardinal thiab \(n\) yog tus lej exponent.
Piv txwv ntawm Cov Teeb Meem Exponent
Lo lus nug thib 1:
Txheeb xyuas tus nqi ntawm \(2^5\).
Kev Sib Tham:
Tus nqi ntawm \(2^5\) yog 2 npaug ntawm nws tus kheej 5 zaug.
\[ 2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 \]
Yog li, tus nqi ntawm \(2^5\) yog 32.
Lo lus nug thib 2:
Xam tus nqi ntawm \( (3^2) \times (3^3) \).
Kev Sib Tham:
Yuav kom daws tau qhov teeb meem no, peb siv tau ib qho ntawm cov cai yooj yim ntawm cov exponents uas hais tias:
\[ a^m \times a^n = a^{m+n} \]
Yog li ntawd,
\[ (3^2) \times (3^3) = 3^{2+3} = 3^5 = 243 \]
Yog li, tus nqi ntawm \( (3^2) \times (3^3) \) yog 243.
Lo lus nug thib 3:
Ua kom yooj yim dua \( \frac{5^6}{5^3} \).
Kev Sib Tham:
Yuav kom yooj yim cov feem pua exponential nrog tib lub hauv paus, peb tuaj yeem siv txoj cai:
\[ \frac{a^m}{a^n} = a^{mn} \]
Yog li ntawd,
\[ \frac{5^6}{5^3} = 5^{6-3} = 5^3 = 125 \]
Yog li, tus nqi ntawm \( \frac{5^6}{5^3} \) yog 125.
Logarithm
Ib qho logarithm yog qhov inverse ntawm ib qho exponent. Feem ntau, yog tias \( a^b = c \), ces \( \log_a c = b \). Hauv lwm lo lus, logarithm ntawm ib tus lej yog qhov exponent uas xav tau los txais tus lej ntawd los ntawm lub hauv paus.
Cov Lus Nug Piv Txwv Logarithm
Lo lus nug thib 4:
Txheeb xyuas tus nqi ntawm \( \log_2 32 \).
Kev Sib Tham:
Yuav kom paub tus nqi ntawm \( \log_2 32 \), peb yuav tsum nrhiav tus nqi ntawm tus exponent uas tsim tau 32 thaum lub hauv paus yog 2.
\[ 2^5 = 32 \]
Txhais tau tias,
\[ \log_2 32 = 5 \]
Yog li, tus nqi ntawm \(\log_2 32 \) yog 5.
Lo lus nug thib 5:
Xam tus nqi ntawm \( \log_3 81 \).
Kev Sib Tham:
Yuav kom paub tus nqi ntawm \( \log_3 81 \), peb yuav tsum nrhiav tus nqi ntawm tus exponent uas tsim tau 81 thaum lub hauv paus yog 3.
\[ 3^4 = 81 \]
Txhais tau tias,
\[ \log_3 81 = 4 \]
Yog li, tus nqi ntawm \(\log_3 81 \) yog 4.
Lo lus nug thib 6:
Ua kom yooj yim rau cov kab lus logarithmic \( \log(100) + \log(10) \).
Kev Sib Tham:
Peb siv tau txoj cai logarithmic uas hais tias:
\[ \log(a) + \log(b) = \log(ab) \]
Yog li ntawd,
\[ \log(100) + \log(10) = \log(100 \times 10) = \log(1000) \]
Peb paub tias 1000 tuaj yeem sau ua \( 10^3 \), yog li:
\[ \log(1000) = \log(10^3) \]
Siv cov cai ntawm logarithms:
\[ \log(10^3) = 3 \]
Yog li, tus nqi ntawm \( \log(100) + \log(10) \) yog 3.
Kev Sib Xyaws ntawm Exponents thiab Logarithms
Qee zaum, cov teeb meem lej xav kom peb siv cov exponents thiab logarithms los daws lawv.
Cov Lus Nug Piv Txwv Ua Ke
Lo lus nug thib 7:
Yog tias \(2^x = 8 \), txiav txim siab tus nqi ntawm x.
Kev Sib Tham:
Yuav kom paub tus nqi ntawm x, peb sau tau 8 ua exponential nrog lub hauv paus 2.
\[ 8 = 2^3 \]
Yog li ntawd, qhov sib npaug ua:
\[ 2^x = 2^3 \]
Vim tias cov hauv paus zoo ib yam, cov exponents yuav tsum zoo ib yam thiab.
\[ x = 3 \]
Yog li, tus nqi ntawm x yog 3.
Lo lus nug thib 8:
Txheeb xyuas tus nqi ntawm \( \log_5 25 \).
Kev Sib Tham:
Yuav kom paub tus nqi ntawm \( \log_5 25 \), peb yuav tsum nrhiav tus nqi ntawm tus exponent uas tsim tau 25 thaum lub hauv paus yog 5.
\[ 5^2 = 25 \]
Txhais tau tias,
\[ \log_5 25 = 2 \]
Yog li, tus nqi ntawm \(\log_5 25 \) yog 2.
Lo lus nug thib 9:
Yog tias \( \log_2 ( x^2 ) = 6 \), txiav txim siab tus nqi ntawm x.
Kev Sib Tham:
Yuav kom txiav txim siab tau tus nqi ntawm x, peb tuaj yeem hloov cov qauv logarithmic mus rau hauv daim ntawv exponential.
\[ \log_2 ( x^2 ) = 6 \]
txhais tau tias,
\[ x^2 = 2^6 \]
\[ x^2 = 64 \]
Yog li, peb yuav tsum nrhiav tus nqi ntawm x uas txaus siab rau \( x^2 = 64 \).
\[ x = 64 \]
\[ x = 8 \]
los yog
\[ x = -8 \]
Yog li, tus nqi ntawm x yog 8 lossis -8.
Xaus
Cov lej qhia thiab cov logarithms yog cov ntsiab lus tseem ceeb hauv kev lej. Los ntawm kev nkag siab thiab kev xyaum ua kom zoo, peb tuaj yeem daws tau ntau yam teeb meem uas cuam tshuam nrog cov lej qhia thiab logarithms. Cov piv txwv saum toj no yuav pab peb nkag siab txog cov ntsiab lus tseem ceeb ntawm cov lej qhia thiab logarithms thiab yuav siv lawv li cas rau kev daws teeb meem. Yog tias peb xyaum ua ntau zaus, peb yuav paub ntau ntxiv thiab txawj ntse hauv kev daws cov teeb meem lej uas cuam tshuam nrog cov lej qhia thiab logarithms.
