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Sagot :
To express [tex]\(\frac{1}{2} \log a + 3 \log b \)[/tex] as a single logarithm and simplify it, follow these steps:
1. Identify the given logarithmic expressions:
[tex]\[ \frac{1}{2} \log(a) \quad \text{and} \quad 3 \log(b). \][/tex]
2. Apply the power rule of logarithms, which states that [tex]\( k \log(x) = \log(x^k) \)[/tex].
For [tex]\(\frac{1}{2} \log(a)\)[/tex], we have:
[tex]\[ \frac{1}{2} \log(a) = \log(a^{\frac{1}{2}}). \][/tex]
For [tex]\(3 \log(b)\)[/tex], we have:
[tex]\[ 3 \log(b) = \log(b^3). \][/tex]
3. Combine the logarithmic expressions using the logarithm addition rule, which states that [tex]\( \log(x) + \log(y) = \log(xy) \)[/tex].
Thus, we combine [tex]\(\log(a^{\frac{1}{2}})\)[/tex] and [tex]\(\log(b^3)\)[/tex]:
[tex]\[ \log(a^{\frac{1}{2}}) + \log(b^3) = \log(a^{\frac{1}{2}} \cdot b^3). \][/tex]
4. Simplify the expression if necessary. In this case, the expression is already simplified.
Therefore, the expression [tex]\(\frac{1}{2} \log a + 3 \log b\)[/tex] as a single logarithm is:
[tex]\[ \boxed{\log(a^{\frac{1}{2}} \cdot b^3)}. \][/tex]
1. Identify the given logarithmic expressions:
[tex]\[ \frac{1}{2} \log(a) \quad \text{and} \quad 3 \log(b). \][/tex]
2. Apply the power rule of logarithms, which states that [tex]\( k \log(x) = \log(x^k) \)[/tex].
For [tex]\(\frac{1}{2} \log(a)\)[/tex], we have:
[tex]\[ \frac{1}{2} \log(a) = \log(a^{\frac{1}{2}}). \][/tex]
For [tex]\(3 \log(b)\)[/tex], we have:
[tex]\[ 3 \log(b) = \log(b^3). \][/tex]
3. Combine the logarithmic expressions using the logarithm addition rule, which states that [tex]\( \log(x) + \log(y) = \log(xy) \)[/tex].
Thus, we combine [tex]\(\log(a^{\frac{1}{2}})\)[/tex] and [tex]\(\log(b^3)\)[/tex]:
[tex]\[ \log(a^{\frac{1}{2}}) + \log(b^3) = \log(a^{\frac{1}{2}} \cdot b^3). \][/tex]
4. Simplify the expression if necessary. In this case, the expression is already simplified.
Therefore, the expression [tex]\(\frac{1}{2} \log a + 3 \log b\)[/tex] as a single logarithm is:
[tex]\[ \boxed{\log(a^{\frac{1}{2}} \cdot b^3)}. \][/tex]
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