Respuesta :

msm555

Answer:

[tex] \log_2 \left(\dfrac{zx^2}{y^2}\right) + \log_9 (y^4x^{12}) [/tex]

Step-by-step explanation:

To simplify the given expression, we can use the properties of logarithms:

[tex] \log_2 z + 2\log_2 x + 4\log_9 y + 12\log_9 x - 2\log_2 y [/tex]

Let's use the properties:

Product Rule: [tex] \log_a mn = \log_a m + \log_a n [/tex]

Power Rule: [tex] n\log_a x = \log_a x^n [/tex]

Quotient Rule: [tex] \log_a \dfrac{m}{n} = \log_a m - \log_a n [/tex]

Applying these rules:

[tex] \log_2 z + \log_2 x^2 + \log_9 y^4 + \log_9 x^{12} - \log_2 y^2 [/tex]

Combine the logarithms using the rules:

[tex] \log_2 (zx^2) + \log_9 (y^4x^{12}) - \log_2 y^2 [/tex]

Now, use the quotient rule for logarithms:

[tex] \log_2 \left(\dfrac{zx^2}{y^2}\right) + \log_9 (y^4x^{12}) [/tex]

Combine the logarithms using the product rule:

[tex] \log_2 \left(\dfrac{zx^2}{y^2}\right) + \log_9 (y^4x^{12}) [/tex]

So, the simplified expression is:

[tex] \Large \boxed{\boxed{ \log_2 \left(\dfrac{zx^2}{y^2}\right) + \log_9 (y^4x^{12})}} [/tex]

semsee45

Answer:

[tex]\log_2\left(\dfrac{zx^2}{y^2}\right)+\log_9(y^4x^{12})[/tex]

Step-by-step explanation:

Given logarithmic expression:

[tex]\log_2(z)+2\log_2(x)+4\log_9(y)+12\log_9(x)-2\log_2(y)[/tex]

To simplify the given expression, we can use the properties of logarithms:

[tex]\boxed{\begin{array}{rl}&\underline{\textsf{Properties of Logarithms}}\\\\\textsf{Product:}&\log_axy=\log_ax + \log_ay\\\\\textsf{Quotient:}&\log_a \left(\dfrac{x}{y}\right)=\log_ax - \log_ay\\\\\textsf{Power:}&\log_ax^n=n\log_ax\end{array}}[/tex]

Begin by collecting the terms with the same base:

[tex]\log_2(z)+2\log_2(x)-2\log_2(y)+4\log_9(y)+12\log_9(x)[/tex]

Apply the power law:

[tex]\log_2(z)+\log_2(x^2)-\log_2(y^2)+\log_9(y^4)+\log_9(x^{12})[/tex]

Now, apply the product law:

[tex]\log_2(zx^2)-\log_2y^2+\log_9(y^4x^{12})[/tex]

Finally, apply the quotient law:

[tex]\log_2\left(\dfrac{zx^2}{y^2}\right)+\log_9(y^4x^{12})[/tex]

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