Pré-cálculo Exemplos

$\log (\sqrt{\frac{3 x^{4} y}{w z^{2}}})$

Etapa 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ para reescrever $\sqrt{\frac{3 x^{4} y}{w z^{2}}}$ como ${(\frac{3 x^{4} y}{w z^{2}})}^{\frac{1}{2}}$ .

$\log ({(\frac{3 x^{4} y}{w z^{2}})}^{\frac{1}{2}})$

Etapa 2

Expanda $\log ({(\frac{3 x^{4} y}{w z^{2}})}^{\frac{1}{2}})$ movendo $\frac{1}{2}$ para fora do logaritmo.

$\frac{1}{2} \log (\frac{3 x^{4} y}{w z^{2}})$

Etapa 3

Reescreva $\log (\frac{3 x^{4} y}{w z^{2}})$ como $\log (3 x^{4} y) - \log (w z^{2})$ .

$\frac{1}{2} (\log (3 x^{4} y) - \log (w z^{2}))$

Etapa 4

Reescreva $\log (3 x^{4} y)$ como $\log (3 x^{4}) + \log (y)$ .

$\frac{1}{2} (\log (3 x^{4}) + \log (y) - \log (w z^{2}))$

Etapa 5

Reescreva $\log (3 x^{4})$ como $\log (3) + \log (x^{4})$ .

$\frac{1}{2} (\log (3) + \log (x^{4}) + \log (y) - \log (w z^{2}))$

Etapa 6

Expanda $\log (x^{4})$ movendo $4$ para fora do logaritmo.

$\frac{1}{2} (\log (3) + 4 \log (x) + \log (y) - \log (w z^{2}))$

Etapa 7

Reescreva $\log (w z^{2})$ como $\log (w) + \log (z^{2})$ .

$\frac{1}{2} (\log (3) + 4 \log (x) + \log (y) - (\log (w) + \log (z^{2})))$

Etapa 8

Expanda $\log (z^{2})$ movendo $2$ para fora do logaritmo.

$\frac{1}{2} (\log (3) + 4 \log (x) + \log (y) - (\log (w) + 2 \log (z)))$

Etapa 9

Simplifique cada termo.

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Etapa 9.1

Aplique a propriedade distributiva.

$\frac{1}{2} (\log (3) + 4 \log (x) + \log (y) - \log (w) - (2 \log (z)))$

Etapa 9.2

Multiplique $2$ por $- 1$ .

$\frac{1}{2} (\log (3) + 4 \log (x) + \log (y) - \log (w) - 2 \log (z))$

Etapa 10

Aplique a propriedade distributiva.

$\frac{1}{2} \log (3) + \frac{1}{2} (4 \log (x)) + \frac{1}{2} \log (y) + \frac{1}{2} (- \log (w)) + \frac{1}{2} (- 2 \log (z))$

Etapa 11

Simplifique.

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Etapa 11.1

Combine $\frac{1}{2}$ e $\log (3)$ .

$\frac{\log (3)}{2} + \frac{1}{2} (4 \log (x)) + \frac{1}{2} \log (y) + \frac{1}{2} (- \log (w)) + \frac{1}{2} (- 2 \log (z))$

Etapa 11.2

Cancele o fator comum de $2$ .

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Etapa 11.2.1

Fatore $2$ de $4 \log (x)$ .

$\frac{\log (3)}{2} + \frac{1}{2} (2 (2 \log (x))) + \frac{1}{2} \log (y) + \frac{1}{2} (- \log (w)) + \frac{1}{2} (- 2 \log (z))$

Etapa 11.2.2

Cancele o fator comum.

$\frac{\log (3)}{2} + \frac{1}{2} (2 (2 \log (x))) + \frac{1}{2} \log (y) + \frac{1}{2} (- \log (w)) + \frac{1}{2} (- 2 \log (z))$

Etapa 11.2.3

Reescreva a expressão.

$\frac{\log (3)}{2} + 2 \log (x) + \frac{1}{2} \log (y) + \frac{1}{2} (- \log (w)) + \frac{1}{2} (- 2 \log (z))$

Etapa 11.3

Combine $\frac{1}{2}$ e $\log (y)$ .

$\frac{\log (3)}{2} + 2 \log (x) + \frac{\log (y)}{2} + \frac{1}{2} (- \log (w)) + \frac{1}{2} (- 2 \log (z))$

Etapa 11.4

Combine $\frac{1}{2}$ e $\log (w)$ .

$\frac{\log (3)}{2} + 2 \log (x) + \frac{\log (y)}{2} - \frac{\log (w)}{2} + \frac{1}{2} (- 2 \log (z))$

Etapa 11.5

Cancele o fator comum de $2$ .

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Etapa 11.5.1

Fatore $2$ de $- 2 \log (z)$ .

$\frac{\log (3)}{2} + 2 \log (x) + \frac{\log (y)}{2} - \frac{\log (w)}{2} + \frac{1}{2} (2 (- \log (z)))$

Etapa 11.5.2

Cancele o fator comum.

$\frac{\log (3)}{2} + 2 \log (x) + \frac{\log (y)}{2} - \frac{\log (w)}{2} + \frac{1}{2} (2 (- \log (z)))$

Etapa 11.5.3

Reescreva a expressão.

$\frac{\log (3)}{2} + 2 \log (x) + \frac{\log (y)}{2} - \frac{\log (w)}{2} - \log (z)$