Question

Find the value of the following logarithms without using a calculator.
​(a) log Subscript 2 Baseline one eighth
​(b) log Subscript 2 Baseline 1
​(c) ln e Superscript 4
​(d) log 0.01
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Part 1
​(a) log Subscript 2 Baseline one eighth
equals
  
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Answer 1

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.