Questions: log3(27) = log5(sqrt(5)) =

Transcript text: $\log _{3}(27)=\quad \log _{5}(\sqrt{5})=$

Solution

Solution Steps

To solve the logarithmic expressions, we need to use the properties of logarithms. For the first expression, we recognize that 27 is a power of 3, specifically $3^3$. For the second expression, we recognize that $\sqrt{5}$ is $5^{1/2}$. We can use the property $\log_b(a^c) = c \cdot \log_b(a)$ to simplify both expressions.

Step 1: Calculate $ \log_{3}(27) $

We know that $ 27 = 3^3 $. Using the property of logarithms, we have: \[ \log_{3}(27) = \log_{3}(3^3) = 3 \cdot \log_{3}(3) = 3 \cdot 1 = 3 \]

Step 2: Calculate $ \log_{5}(\sqrt{5}) $

We recognize that $ \sqrt{5} = 5^{1/2} $. Again, using the property of logarithms, we find: \[ \log_{5}(\sqrt{5}) = \log_{5}(5^{1/2}) = \frac{1}{2} \cdot \log_{5}(5) = \frac{1}{2} \cdot 1 = \frac{1}{2} \]

Final Answer

Thus, the results are: \[ \log_{3}(27) = 3 \quad \text{and} \quad \log_{5}(\sqrt{5}) = \frac{1}{2} \] The final answers are: \[ \boxed{3} \quad \text{and} \quad \boxed{\frac{1}{2}} \]

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