Questions: Evaluate the following expressions. (a) log base 2 of 2^11= (b) log base 3 of 243= (c) log base 4 of 256=

Solution Steps

To evaluate logarithmic expressions, we use the property that $\log_b(a^c) = c \cdot \log_b(a)$. For expressions like $\log_b(b^c)$, the result is simply $c$ because $\log_b(b) = 1$. For $\log_b(a)$ where $a$ is a power of $b$, we find the exponent that makes $b$ equal to $a$.

Using the property of logarithms, we have: $$\log_{2}(2^{11}) = 11$$

Recognizing that $243 = 3^{5}$, we can express the logarithm as: $$\log_{3}(243) = \log_{3}(3^{5}) = 5$$ However, due to numerical precision, the computed value is approximately $4.999999999999999$, which we round to $5$.

Since $256 = 4^{4}$, we find: $$\log_{4}(256) = \log_{4}(4^{4}) = 4$$

Final Answer