Write the logarithm as a sum and/or difference of logarithmic expressions. Eliminate exponents and
radicals, and evaluate logarithms wherever possible. Assume that .

Ask by Greene Schmidt. in the United States

Mar 12,2025

Question

Write the logarithm as a sum and/or difference of logarithmic expressions. Eliminate exponents and
radicals, and evaluate logarithms wherever possible. Assume that .

Ask by Greene Schmidt. in the United States

Mar 12,2025

UpStudy AI · Accepted Answer

To simplify the expression $$ \ln \left(\frac{\sqrt{x^{8}+y}}{e^{3}}\right) $$, we will follow these steps:

1. **Use the property of logarithms for division**: 
   $$
   \ln \left(\frac{a}{b}\right) = \ln(a) - \ln(b)
   $$
   This allows us to separate the logarithm of the numerator and the denominator.

2. **Use the property of logarithms for exponents**: 
   $$
   \ln(a^b) = b \cdot \ln(a)
   $$
   This will help us eliminate the square root.

Now, let's apply these properties step by step.

### Step 1: Apply the division property
$$
\ln \left(\frac{\sqrt{x^{8}+y}}{e^{3}}\right) = \ln(\sqrt{x^{8}+y}) - \ln(e^{3})
$$

### Step 2: Simplify $$\ln(e^{3})$$
Using the property of logarithms for exponents:
$$
\ln(e^{3}) = 3
$$
So we have:
$$
\ln \left(\frac{\sqrt{x^{8}+y}}{e^{3}}\right) = \ln(\sqrt{x^{8}+y}) - 3
$$

### Step 3: Eliminate the square root
Using the property of logarithms for exponents again:
$$
\ln(\sqrt{x^{8}+y}) = \ln((x^{8}+y)^{1/2}) = \frac{1}{2} \ln(x^{8}+y)
$$
Thus, we can rewrite our expression as:
$$
\ln \left(\frac{\sqrt{x^{8}+y}}{e^{3}}\right) = \frac{1}{2} \ln(x^{8}+y) - 3
$$

### Final Expression
The final simplified expression is:
$$
\ln \left(\frac{\sqrt{x^{8}+y}}{e^{3}}\right) = \frac{1}{2} \ln(x^{8}+y) - 3
$$

This is the logarithm expressed as a sum and/or difference of logarithmic expressions, with exponents and radicals eliminated.
$$ \ln \left(\frac{\sqrt{x^{8}+y}}{e^{3}}\right) = \frac{1}{2} \ln(x^{8}+y) - 3 $$

Write the logarithm as a sum and/or difference of logarithmic expressions. Eliminate exponents and
radicals, and evaluate logarithms wherever possible. Assume that .

Upstudy AI Solution

Answer

Solution

Step 1: Apply the division property

Step 2: Simplify

Step 3: Eliminate the square root

Final Expression

The Deep Dive

Related Questions

Latest Pre Algebra Questions

Write the logarithm as a sum and/or difference of logarithmic expressions. Eliminate exponents and radicals, and evaluate logarithms wherever possible. Assume that .

Upstudy AI Solution

Answer

Solution

Step 1: Apply the division property

Step 2: Simplify

Step 3: Eliminate the square root

Final Expression

The Deep Dive

Related Questions

Latest Pre Algebra Questions

Write the logarithm as a sum and/or difference of logarithmic expressions. Eliminate exponents and
radicals, and evaluate logarithms wherever possible. Assume that .