7) Calcule o valor de $$ x $$ usando, em cada caso, as propriedades: $$ \begin{array}{ll} ext { a) } \log x=\log 5+\log 4+\log 3 & ext { b) } 2 \cdot \log x=\log 3+\log 4 \ ext { c) } \log \left(\frac{1}{x} ight)=\log \left(\frac{1}{3} ight)+\log 9 & ext { d) } \frac{1}{2} \cdot \log _{3} x=2 \cdot \log _{3} 10-\log _{3} 4\end{array} $$ 8) Para a compra de uma TV pode-se antar

Question

7) Calcule o valor de $$ x $$ usando, em cada caso, as propriedades: $$ \begin{array}{ll}	ext { a) } \log x=\log 5+\log 4+\log 3 & 	ext { b) } 2 \cdot \log x=\log 3+\log 4 \ 	ext { c) } \log \left(\frac{1}{x}ight)=\log \left(\frac{1}{3}ight)+\log 9 & 	ext { d) } \frac{1}{2} \cdot \log _{3} x=2 \cdot \log _{3} 10-\log _{3} 4\end{array} $$ 8) Para a compra de uma TV pode-se antar

UpStudy AI · Accepted Answer

Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\log_{10}{\left(\frac{1}{x}\right)}=\log_{10}{\left(\frac{1}{3}\right)}+\log_{10}{\left(9\right)}$$
- step1: Find the domain:
  $$\log_{10}{\left(\frac{1}{x}\right)}=\log_{10}{\left(\frac{1}{3}\right)}+\log_{10}{\left(9\right)},x>0$$
- step2: Add the numbers:
  $$\log_{10}{\left(\frac{1}{x}\right)}=\log_{10}{\left(3\right)}$$
- step3: Evaluate the logarithm:
  $$\frac{1}{x}=3$$
- step4: Cross multiply:
  $$1=x\times 3$$
- step5: Simplify the equation:
  $$1=3x$$
- step6: Swap the sides:
  $$3x=1$$
- step7: Divide both sides:
  $$\frac{3x}{3}=\frac{1}{3}$$
- step8: Divide the numbers:
  $$x=\frac{1}{3}$$
- step9: Check if the solution is in the defined range:
  $$x=\frac{1}{3},x>0$$
- step10: Find the intersection:
  $$x=\frac{1}{3}$$
Solve the equation $$ 2 \cdot \log x=\log 3+\log 4 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$2\log_{10}{\left(x\right)}=\log_{10}{\left(3\right)}+\log_{10}{\left(4\right)}$$
- step1: Find the domain:
  $$2\log_{10}{\left(x\right)}=\log_{10}{\left(3\right)}+\log_{10}{\left(4\right)},x>0$$
- step2: Add the terms:
  $$2\log_{10}{\left(x\right)}=\log_{10}{\left(12\right)}$$
- step3: Divide both sides:
  $$\frac{2\log_{10}{\left(x\right)}}{2}=\frac{\log_{10}{\left(12\right)}}{2}$$
- step4: Divide the numbers:
  $$\log_{10}{\left(x\right)}=\frac{\log_{10}{\left(12\right)}}{2}$$
- step5: Convert the logarithm into exponential form:
  $$x=10^{\frac{\log_{10}{\left(12\right)}}{2}}$$
- step6: Evaluate the power:
  $$x=2\sqrt{3}$$
- step7: Check if the solution is in the defined range:
  $$x=2\sqrt{3},x>0$$
- step8: Find the intersection:
  $$x=2\sqrt{3}$$
Solve the equation $$ \frac{1}{2} \cdot \log _{3} x=2 \cdot \log _{3} 10-\log _{3} 4 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\frac{1}{2}\log_{3}{\left(x\right)}=2\log_{3}{\left(10\right)}-\log_{3}{\left(4\right)}$$
- step1: Find the domain:
  $$\frac{1}{2}\log_{3}{\left(x\right)}=2\log_{3}{\left(10\right)}-\log_{3}{\left(4\right)},x>0$$
- step2: Subtract the terms:
  $$\frac{1}{2}\log_{3}{\left(x\right)}=2\log_{3}{\left(5\right)}$$
- step3: Multiply by the reciprocal:
  $$\frac{1}{2}\log_{3}{\left(x\right)}\times 2=2\log_{3}{\left(5\right)}\times 2$$
- step4: Multiply:
  $$\log_{3}{\left(x\right)}=4\log_{3}{\left(5\right)}$$
- step5: Evaluate the logarithm:
  $$x=5^{4}$$
- step6: Evaluate the power:
  $$x=625$$
- step7: Check if the solution is in the defined range:
  $$x=625,x>0$$
- step8: Find the intersection:
  $$x=625$$
Vamos resolver cada um dos itens solicitados:

### a) $$ \log x = \log 5 + \log 4 + \log 3 $$

Usando a propriedade dos logaritmos que diz que $$ \log a + \log b = \log(ab) $$, podemos reescrever a equação:

$$
\log x = \log(5 \cdot 4 \cdot 3)
$$

Calculando o produto:

$$
5 \cdot 4 = 20 \quad \text{e} \quad 20 \cdot 3 = 60
$$

Portanto, temos:

$$
\log x = \log 60
$$

Assim, podemos concluir que:

$$
x = 60
$$

### b) $$ 2 \cdot \log x = \log 3 + \log 4 $$

Novamente, usando a propriedade dos logaritmos:

$$
2 \cdot \log x = \log(3 \cdot 4)
$$

Calculando o produto:

$$
3 \cdot 4 = 12
$$

Assim, temos:

$$
2 \cdot \log x = \log 12
$$

Dividindo ambos os lados por 2:

$$
\log x = \log(12^{1/2}) = \log(2\sqrt{3})
$$

Portanto, concluímos que:

$$
x = 2\sqrt{3}
$$

### c) $$ \log \left(\frac{1}{x}\right) = \log \left(\frac{1}{3}\right) + \log 9 $$

Usando a propriedade dos logaritmos:

$$
\log \left(\frac{1}{x}\right) = \log \left(\frac{1}{3} \cdot 9\right)
$$

Calculando o produto:

$$
\frac{1}{3} \cdot 9 = 3
$$

Assim, temos:

$$
\log \left(\frac{1}{x}\right) = \log 3
$$

Portanto, isso implica que:

$$
\frac{1}{x} = 3 \implies x = \frac{1}{3}
$$

### d) $$ \frac{1}{2} \cdot \log_{3} x = 2 \cdot \log_{3} 10 - \log_{3} 4 $$

Usando a propriedade dos logaritmos:

$$
\frac{1}{2} \cdot \log_{3} x = \log_{3} (10^2) - \log_{3} 4
$$

Isso se torna:

$$
\frac{1}{2} \cdot \log_{3} x = \log_{3} \left(\frac{100}{4}\right)
$$

Calculando a fração:

$$
\frac{100}{4} = 25
$$

Assim, temos:

$$
\frac{1}{2} \cdot \log_{3} x = \log_{3} 25
$$

Multiplicando ambos os lados por 2:

$$
\log_{3} x = 2 \cdot \log_{3} 25
$$

Isso implica que:

$$
x = 25^2 = 625
$$

### Resumo dos resultados:
- a) $$ x = 60 $$
- b) $$ x = 2\sqrt{3} $$
- c) $$ x = \frac{1}{3} $$
- d) $$ x = 625 $$
a) $$ x = 60 $$ b) $$ x = 2\sqrt{3} $$ c) $$ x = \frac{1}{3} $$ d) $$ x = 625 $$

Calcule o valor de usando, em cada caso, as propriedades:

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Upstudy AI Solution

Answer

Solution

a)

b)

c)

d)

Resumo dos resultados:

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Calcule o valor de usando, em cada caso, as propriedades: Para a compra de uma TV pode-se antar

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Answer

Solution

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b)

c)

d)

Resumo dos resultados:

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