Use properties of logarithms to expand the logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible.
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Ask by Macdonald Pena. in the United States

Mar 21,2025

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Use properties of logarithms to expand the logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible.
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Ask by Macdonald Pena. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

1. Begin with the expression  
   $$
   \ln \left[\frac{x^{4} \sqrt{x^{2}+4}}{(x+4)^{4}}\right].
   $$

2. Use the logarithmic property for division:  
   $$
   \ln \left(\frac{A}{B}\right) = \ln(A) - \ln(B).
   $$  
   Thus,  
   $$
   \ln \left[\frac{x^{4} \sqrt{x^{2}+4}}{(x+4)^{4}}\right] = \ln\left(x^4 \sqrt{x^2+4}\right) - \ln\left((x+4)^4\right).
   $$

3. Use the logarithmic property for multiplication in the numerator:  
   $$
   \ln(AB) = \ln(A) + \ln(B).
   $$  
   So,  
   $$
   \ln\left(x^4 \sqrt{x^2+4}\right) = \ln\left(x^4\right) + \ln\left(\sqrt{x^2+4}\right).
   $$

4. Apply the power rule for logarithms:  
   $$
   \ln\left(x^4\right) = 4\ln(x) \quad \text{and} \quad \ln\left((x+4)^4\right) = 4\ln(x+4).
   $$

5. Recall that a square root can be written as a power of one-half:  
   $$
   \sqrt{x^2+4} = (x^2+4)^{\frac{1}{2}},
   $$  
   which implies  
   $$
   \ln\left(\sqrt{x^2+4}\right) = \frac{1}{2}\ln\left(x^2+4\right).
   $$

6. Combine all of the results:  
   $$
   \ln \left[\frac{x^{4} \sqrt{x^{2}+4}}{(x+4)^{4}}\right] = 4\ln(x) + \frac{1}{2}\ln\left(x^2+4\right) - 4\ln(x+4).
   $$

Thus, the fully expanded logarithmic expression is  
$$
\boxed{4\ln(x) + \frac{1}{2}\ln\left(x^2+4\right) - 4\ln(x+4)}.
$$
$$ \ln \left[\frac{x^{4} \sqrt{x^{2}+4}}{(x+4)^{4}}\right] = 4\ln(x) + \frac{1}{2}\ln\left(x^2+4\right) - 4\ln(x+4) $$

Use properties of logarithms to expand the logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible.
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Use properties of logarithms to expand the logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible.
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