Questions: log4(√4) log4(√4)=

Solution Steps

To evaluate the expression \(\log_{4}(\sqrt{4})\) without using a calculator, we can use the properties of logarithms and exponents.

1. Recognize that \(\sqrt{4}\) is the same as \(4^{1/2}\).
2. Use the property of logarithms that \(\log_b(a^c) = c \cdot \log_b(a)\).
3. Since \(\log_{4}(4) = 1\), we can simplify the expression.

We start with the expression \(\sqrt{4}\). We know that: \[ \sqrt{4} = 2 \]

Next, we rewrite the logarithmic expression using the value we found: \[ \log_{4}(\sqrt{4}) = \log_{4}(2) \]

Using the property of logarithms, we can express \(\log_{4}(2)\) in terms of base 4: \[ \log_{4}(2) = \log_{4}(4^{1/2}) = \frac{1}{2} \cdot \log_{4}(4) \] Since \(\log_{4}(4) = 1\), we have: \[ \log_{4}(2) = \frac{1}{2} \cdot 1 = \frac{1}{2} \]

Final Answer