To find the logarithmic values without using a calculator, we'll use the properties of logarithms.
(a) \( \log_2 \left( \frac{1}{8} \right) \)
First, we recognize that \( \frac{1}{8} \) can be rewritten as a power of 2. Specifically, \( 8 = 2^3 \), so \( \frac{1}{8} = 2^{-3} \).
Now, we can rewrite the logarithm:
\[ \log_2 \left( \frac{1}{8} \right) = \log_2 \left( 2^{-3} \right) \]
Using the property of logarithms that states \( \log_b(a^c) = c \cdot \log_b(a) \), we have:
\[ \log_2 \left( 2^{-3} \right) = -3 \cdot \log_2 (2) \]
Since \( \log_2(2) = 1 \):
\[ \log_2 \left( \frac{1}{8} \right) = -3 \cdot 1 = -3 \]
Answer:
\[ \log_2 \left( \frac{1}{8} \right) = -3 \]
(b) \( \log_2(1) \)
The logarithm of 1 in any base is always 0. Thus:
\[ \log_2(1) = 0 \]
Answer:
\[ \log_2(1) = 0 \]
(c) \( \ln(e^4) \)
Applying the property of logarithms \( \ln(a^b) = b \cdot \ln(a) \):
\[ \ln(e^4) = 4 \cdot \ln(e) \]
Since \( \ln(e) = 1 \):
\[ \ln(e^4) = 4 \cdot 1 = 4 \]
Answer:
\[ \ln(e^4) = 4 \]
(d) \( \log(0.01) \)
We can express \( 0.01 \) as a power of 10:
\[ 0.01 = \frac{1}{100} = \frac{1}{10^2} = 10^{-2} \]
Now we apply the property of logarithms:
\[ \log(0.01) = \log(10^{-2}) \]
Using \( \log(a^b) = b \cdot \log(a) \):
\[ \log(10^{-2}) = -2 \cdot \log(10) \]
Since \( \log(10) = 1 \):
\[ \log(0.01) = -2 \cdot 1 = -2 \]
Answer:
\[ \log(0.01) = -2 \]
Thus, the final answers are: (a) -3, (b) 0, (c) 4, (d) -2.
1 answer