Answer :
To solve the expression [tex]\(\frac{1}{3} \log 27 - 2 \log \frac{1}{3}\)[/tex], we will use properties of logarithms and basic operations.
### Step-by-Step Solution:
1. First, let's break down each term in the expression separately:
- For the first term [tex]\(\frac{1}{3} \log 27\)[/tex],
- For the second term [tex]\(2 \log \frac{1}{3}\)[/tex].
2. Evaluate the first part [tex]\(\frac{1}{3} \log 27\)[/tex]:
- Recall that [tex]\(27 = 3^3\)[/tex].
- Using the logarithm property [tex]\(\log(a^b) = b \log a\)[/tex], we can write:
[tex]\[ \log 27 = \log(3^3) = 3 \log 3. \][/tex]
- Now substitute this back into the expression:
[tex]\[ \frac{1}{3} \log 27 = \frac{1}{3} (3 \log 3) = \log 3. \][/tex]
- Numerically, this simplifies to approximately [tex]\( \log 3 \approx 0.47712125471966244\)[/tex].
3. Evaluate the second part [tex]\(2 \log \frac{1}{3}\)[/tex]:
- Using the logarithm property [tex]\(\log \frac{a}{b} = \log a - \log b\)[/tex], we get:
[tex]\[ \log \frac{1}{3} = \log 1 - \log 3. \][/tex]
- Knowing that [tex]\(\log 1 = 0\)[/tex] (since any logarithm at base [tex]\(a\)[/tex] of 1 equals 0), we have:
[tex]\[ \log \frac{1}{3} = 0 - \log 3 = -\log 3. \][/tex]
- Therefore:
[tex]\[ 2 \log \frac{1}{3} = 2 (-\log 3) = -2 \log 3. \][/tex]
- Numerically, this simplifies to approximately [tex]\(-2 \log 3 \approx -0.9542425094393249\)[/tex].
4. Combine the results of both parts:
- Substitute the numerical values back into the original expression:
[tex]\[ \frac{1}{3} \log 27 - 2 \log \frac{1}{3} = (\log 3) - (-2 \log 3 ). \][/tex]
- Simplify the expression:
[tex]\[ = \log 3 + 2 \log 3. \][/tex]
[tex]\[ = 3 \log 3. \][/tex]
- Numerically, this simplifies to approximately:
[tex]\[ 3 \log 3 \approx 1.4313637641589874. \][/tex]
So, the final result of the expression [tex]\(\frac{1}{3} \log 27 - 2 \log \frac{1}{3}\)[/tex] is approximately [tex]\(1.4313637641589874\)[/tex].
### Step-by-Step Solution:
1. First, let's break down each term in the expression separately:
- For the first term [tex]\(\frac{1}{3} \log 27\)[/tex],
- For the second term [tex]\(2 \log \frac{1}{3}\)[/tex].
2. Evaluate the first part [tex]\(\frac{1}{3} \log 27\)[/tex]:
- Recall that [tex]\(27 = 3^3\)[/tex].
- Using the logarithm property [tex]\(\log(a^b) = b \log a\)[/tex], we can write:
[tex]\[ \log 27 = \log(3^3) = 3 \log 3. \][/tex]
- Now substitute this back into the expression:
[tex]\[ \frac{1}{3} \log 27 = \frac{1}{3} (3 \log 3) = \log 3. \][/tex]
- Numerically, this simplifies to approximately [tex]\( \log 3 \approx 0.47712125471966244\)[/tex].
3. Evaluate the second part [tex]\(2 \log \frac{1}{3}\)[/tex]:
- Using the logarithm property [tex]\(\log \frac{a}{b} = \log a - \log b\)[/tex], we get:
[tex]\[ \log \frac{1}{3} = \log 1 - \log 3. \][/tex]
- Knowing that [tex]\(\log 1 = 0\)[/tex] (since any logarithm at base [tex]\(a\)[/tex] of 1 equals 0), we have:
[tex]\[ \log \frac{1}{3} = 0 - \log 3 = -\log 3. \][/tex]
- Therefore:
[tex]\[ 2 \log \frac{1}{3} = 2 (-\log 3) = -2 \log 3. \][/tex]
- Numerically, this simplifies to approximately [tex]\(-2 \log 3 \approx -0.9542425094393249\)[/tex].
4. Combine the results of both parts:
- Substitute the numerical values back into the original expression:
[tex]\[ \frac{1}{3} \log 27 - 2 \log \frac{1}{3} = (\log 3) - (-2 \log 3 ). \][/tex]
- Simplify the expression:
[tex]\[ = \log 3 + 2 \log 3. \][/tex]
[tex]\[ = 3 \log 3. \][/tex]
- Numerically, this simplifies to approximately:
[tex]\[ 3 \log 3 \approx 1.4313637641589874. \][/tex]
So, the final result of the expression [tex]\(\frac{1}{3} \log 27 - 2 \log \frac{1}{3}\)[/tex] is approximately [tex]\(1.4313637641589874\)[/tex].
