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Which of the following expressions is equivalent to the logarithmic expression below?

[tex]\[ \log_4 \frac{8}{x^2} \][/tex]

A. [tex]\( 2 \log_4 8 - \log_4 x \)[/tex]
B. [tex]\( \log_4 8 + 2 \log_4 x \)[/tex]
C. [tex]\( 2 \log_4 8 + \log_4 x \)[/tex]
D. [tex]\( \log_4 8 - 2 \log_4 x \)[/tex]


Sagot :

To solve the logarithmic expression [tex]\(\log_4 \frac{8}{x^2}\)[/tex], we will use properties of logarithms to simplify it step-by-step.

First, we'll apply the logarithmic property dealing with the division inside the logarithm:

[tex]\[ \log_b \left(\frac{m}{n}\right) = \log_b (m) - \log_b (n) \][/tex]

Applying this property, we have:

[tex]\[ \log_4 \left(\frac{8}{x^2}\right) = \log_4 (8) - \log_4 (x^2) \][/tex]

Next, we use the property of logarithms that deals with exponents:

[tex]\[ \log_b (m^n) = n \cdot \log_b (m) \][/tex]

Here, [tex]\( \log_4 (x^2) \)[/tex] can be rewritten using this property:

[tex]\[ \log_4 (x^2) = 2 \cdot \log_4 (x) \][/tex]

Now substituting back into our expression, we get:

[tex]\[ \log_4 (8) - \log_4 (x^2) = \log_4 (8) - 2 \log_4 (x) \][/tex]

Thus, the simplified expression for [tex]\(\log_4 \frac{8}{x^2}\)[/tex] is:

[tex]\[ \log_4 (8) - 2 \log_4 (x) \][/tex]

Comparing this result with the given options:

A. [tex]\(2 \log_4 8 - \log_4 x\)[/tex]

B. [tex]\(\log_4 8 + 2 \log_4 x\)[/tex]

C. [tex]\(2 \log_4 8 + \log_4 x\)[/tex]

D. [tex]\(\log_4 8 - 2 \log_4 x\)[/tex]

We see that option D matches the simplified expression we obtained:

[tex]\[ \log_4 (8) - 2 \log_4 (x) \][/tex]

So, the correct answer is:

D. [tex]\(\log_4 8 - 2 \log_4 x\)[/tex]