Asked by 31
condense the logarithm:
1/3 [2 log(x+3) + 3 log x - log(x^3+1)]
1/3 [2 log(x+3) + 3 log x - log(x^3+1)]
Answers
Answered by
GPT-4o mini
To condense the logarithmic expression
\[
\frac{1}{3} \left[ 2 \log(x+3) + 3 \log x - \log(x^3 + 1) \right],
\]
we can apply the properties of logarithms.
1. **Use the Power Rule**: \( n \log a = \log(a^n) \).
- \( 2 \log(x+3) = \log((x+3)^2) \)
- \( 3 \log x = \log(x^3) \)
This modifies our expression to:
\[
\frac{1}{3} \left[ \log((x+3)^2) + \log(x^3) - \log(x^3 + 1) \right].
\]
2. **Use the Product Rule**: \( \log a + \log b = \log(ab) \).
Combining \( \log((x+3)^2) \) and \( \log(x^3) \):
\[
\frac{1}{3} \left[ \log((x+3)^2 \cdot x^3) - \log(x^3 + 1) \right].
\]
3. **Use the Quotient Rule**: \( \log a - \log b = \log\left(\frac{a}{b}\right) \).
Thus, we can write:
\[
\frac{1}{3} \log\left( \frac{(x+3)^2 \cdot x^3}{x^3 + 1} \right).
\]
4. **Final Condensation**:
This gives us the condensed logarithmic expression:
\[
\log\left( \sqrt[3]{ \frac{(x+3)^2 \cdot x^3}{x^3 + 1} } \right).
\]
So, the final answer is:
\[
\log\left( \sqrt[3]{ \frac{(x+3)^2 \cdot x^3}{x^3 + 1} } \right).
\]
\[
\frac{1}{3} \left[ 2 \log(x+3) + 3 \log x - \log(x^3 + 1) \right],
\]
we can apply the properties of logarithms.
1. **Use the Power Rule**: \( n \log a = \log(a^n) \).
- \( 2 \log(x+3) = \log((x+3)^2) \)
- \( 3 \log x = \log(x^3) \)
This modifies our expression to:
\[
\frac{1}{3} \left[ \log((x+3)^2) + \log(x^3) - \log(x^3 + 1) \right].
\]
2. **Use the Product Rule**: \( \log a + \log b = \log(ab) \).
Combining \( \log((x+3)^2) \) and \( \log(x^3) \):
\[
\frac{1}{3} \left[ \log((x+3)^2 \cdot x^3) - \log(x^3 + 1) \right].
\]
3. **Use the Quotient Rule**: \( \log a - \log b = \log\left(\frac{a}{b}\right) \).
Thus, we can write:
\[
\frac{1}{3} \log\left( \frac{(x+3)^2 \cdot x^3}{x^3 + 1} \right).
\]
4. **Final Condensation**:
This gives us the condensed logarithmic expression:
\[
\log\left( \sqrt[3]{ \frac{(x+3)^2 \cdot x^3}{x^3 + 1} } \right).
\]
So, the final answer is:
\[
\log\left( \sqrt[3]{ \frac{(x+3)^2 \cdot x^3}{x^3 + 1} } \right).
\]
Answered by
GPT-4o mini
Certainly! The condensed form of the logarithmic expression
\[
\frac{1}{3} \left[ 2 \log(x+3) + 3 \log x - \log(x^3 + 1) \right]
\]
is:
\[
\log\left( \sqrt[3]{ \frac{(x+3)^2 \cdot x^3}{x^3 + 1} } \right).
\]
\[
\frac{1}{3} \left[ 2 \log(x+3) + 3 \log x - \log(x^3 + 1) \right]
\]
is:
\[
\log\left( \sqrt[3]{ \frac{(x+3)^2 \cdot x^3}{x^3 + 1} } \right).
\]
Answered by
GPT-4o mini
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