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### Step-by-Step Simplification
Let's start breaking down and simplifying the expression [tex]\(\frac{2}{3} \log 4 + 4 \log 3 + \frac{1}{3} \log 27 - \log 18 - 3 + 3 \log \frac{5}{3}\)[/tex] using properties of logarithms.
1. Expression Rewrite:
[tex]\[ \frac{2}{3} \log 4 + 4 \log 3 + \frac{1}{3} \log 27 - \log 18 - 3 + 3 \log \frac{5}{3} \][/tex]
2. Simplify [tex]\(\frac{1}{3} \log 27\)[/tex]:
[tex]\[ \frac{1}{3} \log 27 = \log (27^{\frac{1}{3}}) = \log 3 \][/tex]
This is because [tex]\( 27 = 3^3 \)[/tex], so [tex]\( 27^{\frac{1}{3}} = 3 \)[/tex].
3. Rewrite with the simplified term:
[tex]\[ \frac{2}{3} \log 4 + 4 \log 3 + \log 3 - \log 18 - 3 + 3 \log \frac{5}{3} \][/tex]
4. Combine the [tex]\(\log 3\)[/tex] terms:
[tex]\[ 4 \log 3 + \log 3 = 5 \log 3 \][/tex]
Now the expression is:
[tex]\[ \frac{2}{3} \log 4 + 5 \log 3 - \log 18 - 3 + 3 \log \frac{5}{3} \][/tex]
5. Combine the logarithms using properties:
- For [tex]\(\log a - \log b\)[/tex]:
[tex]\[ \log \left(\frac{a}{b}\right) \][/tex]
- For [tex]\(n \log a\)[/tex]:
[tex]\[ \log (a^n) \][/tex]
[tex]\[ \frac{2}{3} \log 4 + 5 \log 3 - \log 18 + 3 \log \frac{5}{3} \][/tex]
6. Simplify [tex]\(\frac{2}{3} \log 4\)[/tex]:
[tex]\[ \frac{2}{3} \log 4 = \log (4^{\frac{2}{3}}) \][/tex]
- Find [tex]\(4^{\frac{2}{3}}\)[/tex]:
[tex]\[ 4^{\frac{2}{3}} = (2^2)^{\frac{2}{3}} = 2^{\frac{4}{3}} \][/tex]
So:
[tex]\[ \frac{2}{3} \log 4 = \log (2^{\frac{4}{3}}) \][/tex]
7. Combine all terms:
[tex]\[ \log (2^{\frac{4}{3}}) + 5 \log 3 - \log 18 + 3 \log \frac{5}{3} \][/tex]
8. Simplify [tex]\( \log 18 \)[/tex]:
- [tex]\(18 = 2 \cdot 3^2\)[/tex]:
[tex]\[ \log 18 = \log (2 \cdot 3^2) = \log 2 + 2 \log 3 \][/tex]
9. Rewrite Logarithmic expressions:
[tex]\[ \log (2^{\frac{4}{3}}) + 5 \log 3 - (\log 2 + 2 \log 3) + 3 \log \frac{5}{3} \][/tex]
Combine like terms:
[tex]\[ \log (2^{\frac{4}{3}}) + 5 \log 3 - \log 2 - 2 \log 3 + 3 \log \frac{5}{3} \][/tex]
Simplify:
[tex]\[ \log (2^{\frac{4}{3}}) + 3 \log 3 - \log 2 + 3 \log \frac{5}{3} \][/tex]
10. Simplify [tex]\(3 \log \frac{5}{3}\)[/tex]:
[tex]\[ 3 \log \frac{5}{3} = \log \left(\left(\frac{5}{3}\right)^3\right) = \log \left(\frac{125}{27}\right) \][/tex]
Now the expression is:
[tex]\[ \log (2^{\frac{4}{3}}) + 3 \log 3 - \log 2 + \log \left(\frac{125}{27}\right) \][/tex]
11. Combine and refine the terms:
Use:
[tex]\[ \log a + \log b = \log (a \cdot b) \][/tex]
[tex]\[ \log (2^{\frac{4}{3}}) + 3 \log 3 - \log 2 + \log \left(\frac{125}{27}\right) = \log \left(2^{\frac{4}{3}} \cdot 3^3 \cdot \frac{125}{27} \cdot \frac{1}{2}\right) \][/tex]
12. Combine the logarithms:
Notice [tex]\(3^3 = 27\)[/tex], hence [tex]\(\frac{27}{27} = 1\)[/tex]:
[tex]\[ \log \left(2^{\frac{4}{3}} \cdot \frac{125}{2} \right) \][/tex]
Simplify:
[tex]\[ \log (2^{\frac{4}{3}} \times 62.5) \][/tex]
13. Final Calculation:
The numerical calculation gives:
[tex]\[ (2/3) \cdot \log(4) + 4 \cdot \log(3) + (1/3) \cdot \log(27) - \log(18) + 3 \cdot \log(5/3) - 3 = 2.0593627974889497 \][/tex]
### Conclusion:
The simplified form, considering numerical results, evaluates to approximately [tex]\(2.059\)[/tex] when combined properly.
Let's start breaking down and simplifying the expression [tex]\(\frac{2}{3} \log 4 + 4 \log 3 + \frac{1}{3} \log 27 - \log 18 - 3 + 3 \log \frac{5}{3}\)[/tex] using properties of logarithms.
1. Expression Rewrite:
[tex]\[ \frac{2}{3} \log 4 + 4 \log 3 + \frac{1}{3} \log 27 - \log 18 - 3 + 3 \log \frac{5}{3} \][/tex]
2. Simplify [tex]\(\frac{1}{3} \log 27\)[/tex]:
[tex]\[ \frac{1}{3} \log 27 = \log (27^{\frac{1}{3}}) = \log 3 \][/tex]
This is because [tex]\( 27 = 3^3 \)[/tex], so [tex]\( 27^{\frac{1}{3}} = 3 \)[/tex].
3. Rewrite with the simplified term:
[tex]\[ \frac{2}{3} \log 4 + 4 \log 3 + \log 3 - \log 18 - 3 + 3 \log \frac{5}{3} \][/tex]
4. Combine the [tex]\(\log 3\)[/tex] terms:
[tex]\[ 4 \log 3 + \log 3 = 5 \log 3 \][/tex]
Now the expression is:
[tex]\[ \frac{2}{3} \log 4 + 5 \log 3 - \log 18 - 3 + 3 \log \frac{5}{3} \][/tex]
5. Combine the logarithms using properties:
- For [tex]\(\log a - \log b\)[/tex]:
[tex]\[ \log \left(\frac{a}{b}\right) \][/tex]
- For [tex]\(n \log a\)[/tex]:
[tex]\[ \log (a^n) \][/tex]
[tex]\[ \frac{2}{3} \log 4 + 5 \log 3 - \log 18 + 3 \log \frac{5}{3} \][/tex]
6. Simplify [tex]\(\frac{2}{3} \log 4\)[/tex]:
[tex]\[ \frac{2}{3} \log 4 = \log (4^{\frac{2}{3}}) \][/tex]
- Find [tex]\(4^{\frac{2}{3}}\)[/tex]:
[tex]\[ 4^{\frac{2}{3}} = (2^2)^{\frac{2}{3}} = 2^{\frac{4}{3}} \][/tex]
So:
[tex]\[ \frac{2}{3} \log 4 = \log (2^{\frac{4}{3}}) \][/tex]
7. Combine all terms:
[tex]\[ \log (2^{\frac{4}{3}}) + 5 \log 3 - \log 18 + 3 \log \frac{5}{3} \][/tex]
8. Simplify [tex]\( \log 18 \)[/tex]:
- [tex]\(18 = 2 \cdot 3^2\)[/tex]:
[tex]\[ \log 18 = \log (2 \cdot 3^2) = \log 2 + 2 \log 3 \][/tex]
9. Rewrite Logarithmic expressions:
[tex]\[ \log (2^{\frac{4}{3}}) + 5 \log 3 - (\log 2 + 2 \log 3) + 3 \log \frac{5}{3} \][/tex]
Combine like terms:
[tex]\[ \log (2^{\frac{4}{3}}) + 5 \log 3 - \log 2 - 2 \log 3 + 3 \log \frac{5}{3} \][/tex]
Simplify:
[tex]\[ \log (2^{\frac{4}{3}}) + 3 \log 3 - \log 2 + 3 \log \frac{5}{3} \][/tex]
10. Simplify [tex]\(3 \log \frac{5}{3}\)[/tex]:
[tex]\[ 3 \log \frac{5}{3} = \log \left(\left(\frac{5}{3}\right)^3\right) = \log \left(\frac{125}{27}\right) \][/tex]
Now the expression is:
[tex]\[ \log (2^{\frac{4}{3}}) + 3 \log 3 - \log 2 + \log \left(\frac{125}{27}\right) \][/tex]
11. Combine and refine the terms:
Use:
[tex]\[ \log a + \log b = \log (a \cdot b) \][/tex]
[tex]\[ \log (2^{\frac{4}{3}}) + 3 \log 3 - \log 2 + \log \left(\frac{125}{27}\right) = \log \left(2^{\frac{4}{3}} \cdot 3^3 \cdot \frac{125}{27} \cdot \frac{1}{2}\right) \][/tex]
12. Combine the logarithms:
Notice [tex]\(3^3 = 27\)[/tex], hence [tex]\(\frac{27}{27} = 1\)[/tex]:
[tex]\[ \log \left(2^{\frac{4}{3}} \cdot \frac{125}{2} \right) \][/tex]
Simplify:
[tex]\[ \log (2^{\frac{4}{3}} \times 62.5) \][/tex]
13. Final Calculation:
The numerical calculation gives:
[tex]\[ (2/3) \cdot \log(4) + 4 \cdot \log(3) + (1/3) \cdot \log(27) - \log(18) + 3 \cdot \log(5/3) - 3 = 2.0593627974889497 \][/tex]
### Conclusion:
The simplified form, considering numerical results, evaluates to approximately [tex]\(2.059\)[/tex] when combined properly.
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