Hej czy dałby radę ktoś mi pomóc mam logarytmy i nie ogarniam zadania, zrobiłam całe a i b dalej nie umiem. Poprosiłabym o całe rozpisanie przykładów ❤️klasa 1 liceum
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Logarytmy
Logarytm przy podstawie "a" z liczby "b" daje taką liczbę "c", że liczba "a" podniesiona do potęgi "c" daje liczbę "b".
[tex]\huge\boxed{log_ab=c \to a^c=b}[/tex]
Działania na potęgach
[tex]\huge\boxed{\begin{array}{c|c}a^m*a^n=a^{m+n}\\a^{m}:a^n=a^{m-n}\\a^m*b^m=(a*b)^m\\a^m:b^m=(a:b)^m\\(a^m)^n=a^{m*n}\\a^{\frac1k}=\sqrt[k]a&k\neq0\\a^{\frac{n}k}=\sqrt[k]{a^n}&k\neq0\\a^{-n}=\frac1{a^n}&n\neq0\end{array}}[/tex]
Rozwiązanie:
a)
[tex]log_525=x\\5^x=25\\5^x=5^2\\x=2\\\\\boxed{log_525=2}[/tex]
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[tex]log_7343=x\\7^x=343\\7^x=7^3\\x=3\\\\\boxed{log_7343=3}[/tex]
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[tex]log_61296=x\\6^x=1296\\6^x=6^4\\x=4\\\\\boxed{log_61296=4}[/tex]
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[tex]log_3729=x\\3^x=729\\3^x=3^6\\x=6\\\\\boxed{log_3729=6}[/tex]
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[tex]log_21024=x\\2^x=1024\\2^x=2^{10}\\x=10\\\\\boxed{log_21024=10}[/tex]
____________________________________________________[tex]log_464=x\\4^x=64\\4^x=4^3\\x=3\\\\\boxed{log_464=3}[/tex]
b)
[tex]log_82=x\\8^x=2\\(2^3)^x=2^1\\2^{3x}=2^1\\3x=1 /:3\\x=\frac13\\\\\boxed{log_82=\frac13}[/tex]
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[tex]log_{81}3=x\\81^{x}=3\\(3^4)^x=3\\3^{4x}=3^1\\4x=1 /:4\\x=\frac14\\\\\boxed{log_{81}3=\frac14}[/tex]
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[tex]log_{27}9=x\\27^x=9\\(3^3)^x=3^2\\3^{3x}=3^2\\3x=2 /:3\\x=\frac23\\\\\boxed{log_{27}9=\frac23}[/tex]
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[tex]log_{25}\frac15=x\\25^x=\frac15\\(5^2)^x=5^{-1}\\5^{2x}=5^{-1}\\2x=-1 /:2\\x=-\frac12\\\\\boxed{log_{25}\frac15=-\frac12}[/tex]
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[tex]log69\frac1{69}=x\\69^x=\frac1{69}\\69^x=69^{-1}\\x=-1\\\\\boxed{log_{69}\frac1{69}=-1}[/tex]
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[tex]log_4\frac1{64}=x\\4^x=\frac1{64}\\4^x=64^{-1}\\4^x=(4^3)^{-1}\\4^x=4^{-3}\\x=-3\\\boxed{log_4\frac1{64}=-3}[/tex]
c)
[tex]log_5\sqrt5=x\\5^x=\sqrt5\\5^x=5^{\frac12}\\x=\frac12\\\\\boxed{log_5\sqrt5=\frac12}[/tex]
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[tex]log_3(3\sqrt3)=x\\3^x=3\sqrt3\\3^x=3*3^{\frac12}\\3^x=3^{1+\frac12}\\3^x=3^{\frac32}\\x=\frac32\\\\\boxed{log_3(3\sqrt3)=\frac32}[/tex]
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[tex]log_2(4\sqrt[3]{16})=x\\2^x=4\sqrt[3]{16}\\2^x=2^2*16^{\frac13}\\2^x=2^2*(2^4)^{\frac13}\\2^x=2^{2}*2^{\frac43}\\2^x=2^{2+\frac43}\\2^x=2^{\frac{10}3}\\x=\frac{10}3\\\\\boxed{log_2(4\sqrt[3]{16})=\frac{10}3}[/tex]
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[tex]log_3(\frac13\sqrt[3]3)=x\\3^x=\frac13\sqrt[3]3\\3^x=3^{-1}*3^{\frac13}\\3^x=3^{-1+\frac13}\\3^x=3^{-\frac23}\\x=-\frac23\\\\\boxed{log_3(\frac13\sqrt[3]3)=-\frac23}[/tex]
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[tex]log_2(0,125\sqrt[6]{64})=x\\2^x=0,125\sqrt[6]{64}\\2^x=\frac{125}{1000}*64^{\frac16}\\2^x=\frac18*(2^6)^{\frac16}\\2^x=2^{-3}*2^1\\2^x=2^{-3+1}\\2^x=2^{-2}\\x=-2\\\\\boxed{log_2(0,125\sqrt[6]{64})=-2}[/tex]
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[tex]log\sqrt{0,001}=x\\10^x=\sqrt{0,001}\\10^x=(\frac{1}{1000})^{\frac12}\\10^x=(10^{-3})^{\frac12}\\10^x=10^{-\frac32}\\x=-\frac32\\\\\boxed{log\sqrt{0,001}=-\frac32}[/tex]
d)
[tex]log_{\sqrt2}32=x\\(\sqrt{2})^x=32\\(2^{\frac12})^x=2^5\\2^{\frac12x}=2^5\\\frac12x=5 /*2\\x=10\\\\\boxed{log_{\sqrt{2}}32=10}[/tex]
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[tex]log_{\sqrt6}\frac1{36}=x\\(\sqrt6)^x=\frac1{36}\\(6^{\frac12})^x=36^{-1}\\6^{\frac12x}=(6^2)^{-1}\\6^{\frac12x}=6^{-2}\\\frac12x=-2/*2\\x=-4\\\\\boxed{log_{\sqrt6}\frac1{16}=-4}}[/tex]
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[tex]log_{\sqrt[3]3}27=x\\(\sqrt[3]3)^x=27\\(3^{\frac13})^x=3^3\\3^{\frac13x}=3^3\\\frac13x=3 /*3\\x=9\\\\\boxed{log_{\sqrt[3]3}27=9}[/tex]
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[tex]log_{2\sqrt2}16=x\\(2\sqrt2)^x=16\\(2*2^{\frac12})^x=16\\(2^{\frac32})^x=2^{4}\\2^{\frac32x}=2^x\\\frac32x=4/*\frac{2}3\\x=\frac83\\\\\boxed{log_{2\sqrt2}16=\frac83}[/tex]
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[tex]log_{27\sqrt[4]3}9=x\\(27\sqrt[4]3)^x=9\\(3^3*3^{\frac14})^x=3^2\\(3^{\frac{13}4})^x=3^2\\3^{\frac{13}4x}=3^2\\\frac{13}4x=2 /*4\\13x=8 /:13\\x=\frac8{13}\\\\\boxed{log_{27\sqrt[4]3}9=\frac8{13}}[/tex]
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[tex]log_{7\sqrt[4]{49}}\frac17=x\\(7\sqrt[4]{49})^x=\frac17\\(7*49^{\frac14})^x=7^{-1}\\(7*(7^2)^{\frac14})^x=7^{-1}\\(7*7^{\frac12})^x=7^{-1}\\(7^{\frac32})^x=7^{-1}\\7^{\frac32x}=7^{-1}\\\frac32x=-1 /*\frac23\\x=-\frac23\\\\\boxed{log_{7\sqrt[4]{49}}\frac17=-\frac23}[/tex]
e)
[tex]log_{0,2}0,04=x\\0,2^x=0,04\\(\frac2{10})^x=\frac4{100}\\(\frac2{10})^x=(\frac2{10})^2\\x=2\\\\\boxed{log_{0,2}0,04=2}[/tex]
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[tex]log_{0,25}0,5=x\\0,25^x=0,5\\(\frac14)^x=\frac12\\((\frac12)^2)^x=\frac12\\(\frac12)^{2x}=\frac12\\2x=1 /:2\\x=\frac12\\\\\boxed{log_{0,25}0,5=\frac12}[/tex]
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[tex]log_{0,2}0,008=x\\0,2^x=0,008\\(\frac2{10})^x=\frac8{1000}\\(\frac2{10})^x=(\frac2{10})^3\\x=3\\\\\boxed{log_{0,2}0,008=3}[/tex]
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[tex]log_{0,5}0,125=x\\0,5^x=0,125\\(\frac12)^x=\frac18\\(\frac12)^x=(\frac12)^3\\x=3\\\\\boxed{log_{0,5}0,125=3}[/tex]
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[tex]log_{0,0016}5=x\\0,0016^x=5\\(\frac{16}{10000})^x=5\\(\frac1{625})^x=5\\(625^{-1})^x=5\\(5^4)^{-x}=5\\5^{-4x}=5\\-4x=1 /:(-4)\\x=-\frac14\\\\\boxed{log_{0,0016}5=-\frac14}[/tex]
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[tex]log_{0,11}0,0121=x\\0,11^x=0,0121\\(\frac{11}{100})^x=\frac{121}{10000}\\(\frac{11}{100})^x=(\frac{11}{100})^2\\x=2\\\\\boxed{log_{0,11}0,0121=2}[/tex]