Question

Considerando log de 2 na base 3 = a e log de 5 na base 3 = b, é correto afirmar que
01. log de 162 na base 3 = a + 4.
02. log de raiz de 75 na base 3 = b/2.
04. log de 12 na base 15 = (1+a)/(1+b).
08. o valor de X na equação 5^x = 10 é a/b +1.
16. log de 72 na base 10 = (3a)/(a+b).

Soma: ____

Answer 1

Explicação passo-a-passo:

$log_{3}(2) = a \\ log_{3}(5) = b$

01)

$log_{3}(162) = log_{3}(2. {9}^{2} )$

$log_{3}( {2.3}^{4} ) = log_{3}(2) + 4 log_{3}(3)$

$a + 4 log_{3}(3) = a + 4$

02)

$log_{3}( {75}^{ \frac{1}{2} } ) = \frac{1}{2} log_{3}( {5}^{3} )$

$\frac{3}{2} log_{3}(5) = \frac{3}{2} b$

04)

$log_{15}(12) = \frac{ log_{3}(15) }{ log_{3}(12) } = log_{3}(15) - log_{3}(12)$

$\frac{ log_{3}(3.5) }{ log_{3}( {2}^{2}.3 ) } = \frac{ log_{3}(3) + log_{3}(5) }{ 2log_{3}(2) + log_{3}(3) }$

$\frac{1 + b}{2a + 1}$

08)

${5}^{x} = 10$

$log_{5}(10) = x$

$\frac{ log_{3}(5) }{ log_{3}(10) } = x$

$\frac{b}{ log_{3}(2) + log_{3}(5) } = x$

$\frac{b}{a + b} = x$

$\frac{b}{a} + 1 = x$

16)

$log_{10}(72) = \frac{ log_{3}(10) }{ log_{3}(72) } = \frac{ log_{3}(2.5) }{ log_{3}( {2}^{3} . {3}^{2} ) }$

$\frac{ log_{3}(2) + log_{3}(3) }{ log_{ 3}( {2}^{3}) + log_{3}( {3}^{2} ) }$

$\frac{a + b}{3a + 2}$

Soma: 01+04=05