29-Logaritmos Explora\303\247\303\243o dos logaritmos - seu conceito, rela\303\247\303\243o com equa\303\247\303\265es esponenciais, logaritmos comuns e naturais, computar outros logaritmos, simplificar express\303\265es logar\303\255tmicas, solucionar equa\303\247\303\265es logar\303\255tmicas e os gr\303\241ficos de fun\303\247\303\265es logar\303\255tmicas.

restart; with(plots):

Warning, the name changecoords has been redefined

txln := `\134n____________________________________________________________________________`:

Quando lidamos com n\303\272meros que s\303\243o diferentes pot\303\252ncias da mesma base: a^k $ k = 1..12;

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Faz mais sentido lidarmos apenas com os expoentes por vez. Estes s\303\243o logaritmos das express\303\265es originais. k $ k = 1..12: `expoente `= %;

NiMvJSpleHBvZW50ZX5HNi4iIiIiIiMiIiQiIiUiIiYiIiciIigiIikiIioiIzUiIzYiIzc=

'' log[a](a^k)' = k ' $ k = 1..12;

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'' log[a](a^k)' = k ' $ k = 1..12;

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Pot\303\252ncias negativas significam expoentes negativos. a^(-k) $ k = 1..10: `Valores de 7 ` = %; (-k) $ k = 1..10: `Logs desse n\303\272meros (base 7) ` = %;

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7^k $ k = 1..10: `Valores de 7 ` = %; simplify(log[7](7^k)) $ k = 1..10: `Logs desses n\303\272meros (base 7) ` = %;

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7^(-k) $ k = 1..10: `valores negativos de 7 ` = %; simplify(log[7](7^(-k))) $ k = 1..10: `Logs desses n\303\272meros (base 7) ` = %;

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O logaritmo \303\251 o expoente - quando voc\303\252 pega um n\303\272mero de base e o eleva a um expoente. Portanto, cada logaritmo depende de dois n\303\272meros - a base e o resultado. expoente := 7; base := 19; `base^expoente` = base^exponente; `log do resultado `= round(log[19](base^exponente)); `Isto \303\251 o mesmo !`;

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expoente := 4.5; base := 3; `base^expoente` = base^exponente; `log do resultado `= log[3](base^exponente); `Isto \303\251 o mesmo!`;

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expoente := 3.52; 6^exponente; `log do resultado `= log[6](%); `Isto \303\251 o mesmo!`;

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expoente := 11.0892; 134^exponente; `log do resultado `= log[134](%); `Isto \303\251 o mesmo!`;

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Como percebemos acima, o logar\303\255tmo \303\251 dependente de dois n\303\272meros - da base e do resultado. O n\303\272mero resultante tem dois logaritmo deiferentes relativos a duas bases diferentes. Result := 64: for k from 1 to 12 do base:=2^k: print(cat(`O logaritmo de 64 (base `, base, `) \303\251 `,simplify(log[base](64)))); od:

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Uma das transforma\303\247\303\265es mais fundamentais \303\251 entre equa\303\247\303\265es logar\303\255tmicas e equa\303\247\303\265es exponenciais. Certas vezes precisamos converter uma para a outra ou vice-versa. Essas duas equa\303\247\303\265es s\303\243o equivalentes \303\240s mesmas letras que representam os mesmos valores: a^b = c;

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log[a](c) = b;

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Essas duas equa\303\247\303\265es s\303\243o equivalentes \303\240s mesmas letras que representam os mesmos valores: "2^3" = 8; 'log[2](8)' = 3;

NiMvUSQyXjM2IiIiKQ==NiMvLSYlJGxvZ0c2IyIiIzYjIiIpIiIk

`Note que as duas equa\303\247\303\265es tem a mesma solu\303\247\303\243o!`; 'log[3](7)' = c; solve(%,c): % = evalf(%,20); 3^c = 7; solve(%,c): % = evalf(%,20);

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`Note que as duas equa\303\247\303\265es tem a mesma solu\303\247\303\243o!`; 'log[12](y)' = 3; solve(%,y); 12^3 = y; solve(%,y);

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Existem dois casos especiais de logaritmos que s\303\243o usados comumente: - quando um logaritmo tem base 10, \303\251 chamado de logaritmo comum. Frequentemente, utilizamos a nota\303\247\303\243o log a (sem uma base aparente) para representar um logaritmo comum. - quando um logaritmo tem uma base e, \303\251 chamado de logar\303\255tmo natural. Usamos a nota\303\247\303\243o ln a para representar os logaritmos naturais. O Maple tem fun\303\247\303\265es especiais para cada um dos casos. Esse \303\251 um logaritmo comum - frequentemente usado em aulas de ci\303\252ncias . `logaritmo comum`; 'log[10](10^13)': '%' = simplify(%); 'log10(10^13)': '%' = simplify(%);

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`logaritmo comum`; 'log[10](231)': '%' = evalf(simplify(%)); 'log10(231)': '%' = evalf(simplify(%));

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`logaritmo comum`; 'log[10](.032)': '%' = evalf(simplify(%)); 'log10(.032)': '%' = evalf(simplify(%));

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Este \303\251 um logaritmo natural, naturalmente usado em matem\303\241tica. `logaritmo natural`; 'log[exp(1)](31)': '%' = evalf(simplify(%)); 'ln(31)': '%' = evalf(simplify(%));

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`logaritmo natural`; 'log[exp(1)](10)': '%' = evalf(simplify(%)); 'ln(10)': '%' = evalf(simplify(%));

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`logaritmo natural`; 'log[exp(1)](.01)': '%' = evalf(simplify(%)); 'ln(.01)': '%' = evalf(simplify(%));

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H\303\241 v\303\241rias regras para logaritmos. Vejamos algumas delas e alguns exemplos. print(cat(txln,`\134n1. Produtos`)); 'log[b](p*q) = log[b](p) + log[b](q)';print(txln);

NiMlZHB8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198KzEuflByb2R1dG9zRw==NiMvLSYlJGxvZ0c2IyUiYkc2IyomJSJwRyIiIiUicUciIiIsJi0mJSRsb2dHNiMlImJHNiMlInBHIiIiLSYlJGxvZ0c2IyUiYkc2IyUicUciIiI=NiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

Perceba que temos as mesmas respostas para ambos os casos! ''log[7](13*41)'': % = evalf(eval(%)); ''log[7](13) +log[7](41)'': % = evalf(eval(%));

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print(cat(txln,`\134n2. Quocientes`)); 'log[b](p/q)' = 'log[b](p) - log[b](q)'; ;print(txln);

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''log[7](13/41)'': % = evalf(eval(%)); ''log[7](13) - log[7](41)'': % = evalf(eval(%));

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print(cat(txln,`\134n3. resposta zero \303\251 um`));'log[b](1)'=0;print(txln);

NiMlYHF8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198KzMufnJlc3Bvc3Rhfnplcm9+fGp5fGp2fGh2fnVtRw==NiMvLSYlJGxvZ0c2IyUiYkc2IyIiIiIiIQ==NiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

''log[5](1)'': % = simplify(eval(%)); ''log[22](1)'': % = simplify(eval(%)); ''log[53](1)'': % = simplify(eval(%));

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print(cat(txln,`\134n4. Resposta um \303\251 a identidade`));'log[b](b)' = 1;print(txln);

NiMlaHF8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198KzQuflJlc3Bvc3RhfnVtfnxqeXxqdnxodn5hfmlkZW50aWRhZGVHNiMvLSYlJGxvZ0c2IyUiYkc2IyUiYkciIiI=NiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

''log[5](5)'': % = simplify(eval(%)); ''log[22](22)'': % = simplify(eval(%)); ''log[53](53)'': % = simplify(eval(%));

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print(cat(txln,`\134n5. Respostas - mesmas bases`));'log[b](b^r) = r';print(txln);

NiMlZHF8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198KzUuflJlc3Bvc3Rhc34tfm1lc21hc35iYXNlc0c=NiMvLSYlJGxvZ0c2IyUiYkc2IyklImJHJSJyRyUickc=NiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

''log[4.71](4.71^3.22)'': % = evalf(eval(%)); 3.22;

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print(cat(txln,`\134n6. Respostas - bases diferentes`)); 'log[b](p^r) = r*log[b](p)';print(txln);

NiMlaHF8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198KzYuflJlc3Bvc3Rhc34tfmJhc2VzfmRpZmVyZW50ZXNHNiMvLSYlJGxvZ0c2IyUiYkc2IyklInBHJSJyRyomJSJyRyIiIi0mJSRsb2dHNiMlImJHNiMlInBHIiIiNiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

''log[2](4.71^3.22)'': % = evalf(eval(%)); ''3.22 * log[2](4.71)'': % = evalf(eval(%));

NiMvLSYlJGxvZ0c2IyIiIzYjJCIrIylwTnA5ISIoJCIrTDYvKj4oISIqNiMvLCQqJiQiJEEkISIjIiIiLSYlJGxvZ0c2IyIiIzYjJCIkciUhIiMiIiIiIiIkIitMNi8qPighIio=

print(cat(txln,`\134n7. Log Aplicado ao Expoente`)); 'b^log[b](a) = a';print(txln);

NiMlZHF8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198KzcufkxvZ35BcGxpY2Fkb35hb35FeHBvZW50ZUc=NiMvKSUiYkctJiUkbG9nRzYjJSJiRzYjJSJhRyUiYUc=NiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

''9^log[9](7)'': % = evalf(eval(%)); 7;

NiMvKSIiKi0mJSRsb2dHNiMiIio2IyIiKCQiKy0rKytxISIqNiMiIig=

print(cat(txln,`\134nNote.......Soma : ISTO N\303\203O \303\211 UMA REGRA - Isto \303\251 um ERRO comum !`)); 'log[b](p + q) = log[b](p + q)'; ;print(txln);

NiMlXnR8K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX198K05vdGUuLi4uLi4uU29tYX46fklTVE9+TnxqeXxqdnxodk9+fGp5fGp2fGh2flVNQX5SRUdSQX4tfklzdG9+fGp5fGp2fGh2fnVtfkVSUk9+Y29tdW1+IUc=NiMvLSYlJGxvZ0c2IyUiYkc2IywmJSJwRyIiIiUicUciIiItJiUkbG9nRzYjJSJiRzYjLCYlInBHIiIiJSJxRyIiIg==NiMlaG98K19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19fX19H

''log[7](13 + 41)'': % = evalf(eval(%)); ''log[7](13) + log[7](41)'': % = evalf(eval(%)); `N\303\203O \303\251 igual !`;

NiMvLSYlJGxvZ0c2IyIiKDYjIiNhJCIrIUhLKlw/ISIqNiMvLCYtJiUkbG9nRzYjIiIoNiMiIzgiIiItJiUkbG9nRzYjIiIoNiMiI1QiIiIkIitgPF9FSyEiKg==NiMlMk58anl8anZ8aHZPfnxqeXxqdnxodn5pZ3VhbH4hRw==

''log[7](92 + 37)'': % = evalf(eval(%)); ''log[7](92) + log[7](37)'': % = evalf(eval(%)); `N\303\203O \303\251 igual !`;

NiMvLSYlJGxvZ0c2IyIiKDYjIiRIIiQiK1ImXHVcIyEiKg==NiMvLCYtJiUkbG9nRzYjIiIoNiMiIyMqIiIiLSYlJGxvZ0c2IyIiKDYjIiNQIiIiJCIrJVElUXpUISIqNiMlMk58anl8anZ8aHZPfnxqeXxqdnxodn5pZ3VhbH4hRw==

Observe essa "coincid\303\252ncia": 'log[2](3)' = evalf( log[2](3), 20);

NiMvLSYlJGxvZ0c2IyIiIzYjIiIkJCI1Oj1jNnMrRCdcZSIhIz4=

ln(3)/ln(2): % = evalf(%, 20);

NiMvKiYtJSNsbkc2IyIiJCIiIi0lI2xuRzYjIiIjISIiJCI1Oj1jNnMrRCdcZSIhIz4=

Esse \303\251 um exemplo. expoente := evalf(ln(7)/ln(5),20);

NiM+JSlleHBvZW50ZUckIjVvYm5AN2I+MTQ3ISM+

evalf( log[5](7), 20 );

NiMkIjVvYm5AN2I+MTQ3ISM+

round( 5^exponent );

NiMtJSZyb3VuZEc2IykiIiYlKWV4cG9uZW50Rw==

Observe que esses tr\303\252s t\303\252m o mesmo valor; log[a](b); log10(b)/log10(a);ln(b)/ln(a);

NiMqJi0lI2xuRzYjJSJiRyIiIi0lI2xuRzYjJSJhRyEiIg==NiMqJi0lI2xuRzYjJSJiRyIiIi0lI2xuRzYjJSJhRyEiIg==NiMqJi0lI2xuRzYjJSJiRyIiIi0lI2xuRzYjJSJhRyEiIg==

Esse \303\251 um m\303\251todo que voc\303\252 pode usar para calcular logaritmos em uma calculadora - quando n\303\243o s\303\243o logaritmos comuns nem naturais. 'log[3](10)'; % = evalf(%);

NiMtJiUkbG9nRzYjIiIkNiMiIzU=NiMvKiYtJSNsbkc2IyIjNSIiIi0lI2xuRzYjIiIkISIiJCIrdUshZjQjISIq

Usando essas regras, podemos simplificar express\303\265es com logaritmos. log( x^2 * y^10 /z^3);

NiMtJSNsbkc2IyooKSUieEciIiMiIiIpJSJ5RyIjNSIiIiklInpHIiIkISIi

log( x^2 ) + log( y^10) - log( z^3);

NiMsKC0lI2xuRzYjKiQpJSJ4RyIiIyIiIiIiIi0lI2xuRzYjKiQpJSJ5RyIjNSIiIiIiIi0lI2xuRzYjKiQpJSJ6RyIiJCIiIiEiIg==

2*log(x) + 10*log(y) -3*log(z);

NiMsKComIiIjIiIiLSUjbG5HNiMlInhHIiIiIiIiKiYiIzUiIiItJSNsbkc2IyUieUciIiIiIiIqJiIiJCIiIi0lI2xuRzYjJSJ6RyIiIiEiIg==

N\303\263s mesmos fizemos todo o trabalho. No entanto, podemos usar o Maple para verificar se a resposta est\303\241 certa. Se escolhermos dois n\303\272meros aleat\303\263reos, deveremos obter os mesmos resultados na express\303\243o original e da nossa express\303\243o simplificada final. Isso n\303\243o quer dizer que a resposta est\303\241 certa, mas j\303\241 \303\251 um bom indicativo. Se as duas express\303\265es estiverem de acordo com as diversas substitui\303\247\303\265es, podemos ter certeza de que est\303\241 correto. Se uma delas n\303\243o concordar, teremos certeza de que a resposta n\303\243o est\303\241 certa! `\134nConferindo .... \134n`; evalf( subs( {x = 2, y = 3, z = 7}, log( x^2 * y^10 /z^3) )); evalf( subs( {x = 2, y = 3, z = 7}, 2*log(x) + 10*log(y) -3*log(z)));

NiMlM3wrQ29uZmVyaW5kb34uLi4ufnwrRw==NiMkIissb29NbCEiKg==NiMkIisub29NbCEiKg==

`\134nConferindo .... \134n`; evalf( subs( {x = 10, y = 22, z = 47}, log( x^2 * y^10 /z^3) )); evalf( subs( {x = 10, y = 22, z = 47}, 2*log(x) + 10*log(y) -3*log(z)));

NiMlM3wrQ29uZmVyaW5kb34uLi4ufnwrRw==NiMkIisiPjpsUiMhIik=NiMkIisiPjpsUiMhIik=

Vejamos outros exemplos. log( sqrt( (a+b)/c^5 ));

NiMtJSNsbkc2IyokKSomLCYlImFHIiIiJSJiRyIiIiIiIiklImNHIiImISIiIyIiIiIiIyIiIg==

log( ( (a+b)/c^5 )^(1/2) );

NiMtJSNsbkc2IyokKSomLCYlImFHIiIiJSJiRyIiIiIiIiklImNHIiImISIiIyIiIiIiIyIiIg==

(1/2)*log( (a+b)/c^5 );

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(1/2)*( log(a+b) - 5*log(c) );

NiMsJiomIyIiIiIiIyIiIi0lI2xuRzYjLCYlImFHIiIiJSJiRyIiIiIiIiIiIiomIyIiJiIiIyIiIi0lI2xuRzYjJSJjRyIiIiEiIg==

Vamos verificar essa resposta tamb\303\251m. `\134nConferindo .... \134n`; evalf( subs( {a = 2, b = 3, c = 7}, log( sqrt( (a+b)/c^5 )) )); evalf( subs( {a = 2, b = 3, c = 7}, (1/2)*( log(a+b) - 5*log(c) )));

NiMlM3wrQ29uZmVyaW5kb34uLi4ufnwrRw==NiMkISs8azBnUyEiKg==NiMkISs7azBnUyEiKg==

`\134nConferindo .... \134n`; evalf( subs( {a = 31, b = 4, c = 9}, log( sqrt( (a+b)/c^5 )) )); evalf( subs( {a = 31, b = 4, c = 9}, (1/2)*( log(a+b) - 5*log(c) )));

NiMlM3wrQ29uZmVyaW5kb34uLi4ufnwrRw==NiMkISs4dVE6UCEiKg==NiMkISs3dVE6UCEiKg==

Esses s\303\243o alguns dos gr\303\241ficos b\303\241sicos plot( ln(x), x = 0..100);

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Esses s\303\243o os gr\303\241ficos de log[2] ...log[10] no mesmo gr\303\241fico. Observe que eles todos passam pelo mesmo ponto (1,0) e t\303\252m a mesma vertical ass\303\255mptota. plot( { seq(log[k](x) , k = 2..10)} , x = 0..10);

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Esses s\303\243o g\303\241rficos de logaritmos deslocados para a direita. plot( { seq(ln(x - k) , k = 0..10)} , x = 0..30, y = -4..3);

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Esses s\303\243o g\303\241rficos de logaritmos deslocados para a cima. plot( { seq(ln(x) + k , k = -6..6)} , x = 0..20);

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Esses s\303\243o g\303\241rficos de logaritmos deslocados 4 para a direita, e 3 para cima. plot( ln(x - 4) + 3 , x = 0..20);

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Podemos observar melhor esse deslocamento ao plotarmos onde os eixos se encontrar\303\243o no deslocamento. plots[display]( plot( ln(x - 4) + 3 , x = 0..20), plot( [[4,-1],[4,6]], color = blue, linestyle = 2), plot( [[0,3],[20,3]], color = blue, linestyle = 2));

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

Exemplo 1 : 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. log[7]( 2*x + 1) = 3;

NiMvKiYtJSNsbkc2IywmKiYiIiMiIiIlInhHIiIiIiIiIiIiIiIiIiIiLSUjbG5HNiMiIighIiIiIiQ=

solve( %, x);

NiMiJHIi

7^3 = 2*x + 1; solve( %,x);

NiMvIiRWJCwmKiYiIiMiIiIlInhHIiIiIiIiIiIiIiIiNiMiJHIi

Exemplo 2 : 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. log[4]( x + 2) + log[4](x - 2) = 2;

NiMvLCYqJi0lI2xuRzYjLCYlInhHIiIiIiIjIiIiIiIiLSUjbG5HNiMiIiUhIiIiIiIqJi0lI2xuRzYjLCYlInhHIiIiIiIjISIiIiIiLSUjbG5HNiMiIiUhIiIiIiIiIiM=

solve(%, x);

NiMsJComIiIjIiIiKSIiJiMiIiIiIiMiIiIiIiI=

log[4]( (x+2)*(x-2) ) = 2;

NiMvKiYtJSNsbkc2IyomLCYlInhHIiIiIiIjIiIiIiIiLCYlInhHIiIiIiIjISIiIiIiIiIiLSUjbG5HNiMiIiUhIiIiIiM=

(x+2)*(x-2) = 4^2; expand(%); lhs(%)-rhs(%); solve(%);

NiMvKiYsJiUieEciIiIiIiMiIiIiIiIsJiUieEciIiIiIiMhIiIiIiIiIzs=NiMvLCYqJCklInhHIiIjIiIiIiIiIiIlISIiIiM7NiMsJiokKSUieEciIiMiIiIiIiIiIz8hIiI=NiQsJComIiIjIiIiKSIiJiNGJkYlRiZGJiwkRiQhIiI=