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

Transcript text: Evaluate the following expressions. (a) $\log _{2} 2^{11}=$ (b) $\log _{3} 243=$ (c) $\log _{4} 256=$

Solution

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 $a$.

Step 1: Evaluate $ \log_{2}(2^{11}) $

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

Step 2: Evaluate $ \log_{3}(243) $

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 $.

Step 3: Evaluate $ \log_{4}(256) $

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