Question: evaulate without using a calculator a) log 81 base 7 x log 100 base 3 x lg 49 f) log 16 base x times log x base 64
1. **Problem Statement:**
Evaluate without using a calculator:
(a) $\log_7 81 \times \log_3 100 \times \lg 49$
(f) $\log_x 16 \times \log_{64} x$
2. **Recall the change of base formula:**
$$\log_a b = \frac{\log_c b}{\log_c a}$$
for any positive base $c \neq 1$.
3. **Evaluate (a):**
- Express each term in terms of base 10 logarithms (common logs, $\lg$):
$$\log_7 81 = \frac{\lg 81}{\lg 7}, \quad \log_3 100 = \frac{\lg 100}{\lg 3}, \quad \lg 49 = \lg 49$$
- Substitute:
$$\log_7 81 \times \log_3 100 \times \lg 49 = \frac{\lg 81}{\lg 7} \times \frac{\lg 100}{\lg 3} \times \lg 49$$
- Write numbers as powers:
$$81 = 3^4, \quad 100 = 10^2, \quad 49 = 7^2$$
- Replace:
$$= \frac{\lg (3^4)}{\lg 7} \times \frac{\lg (10^2)}{\lg 3} \times \lg (7^2)$$
- Use log power rule $\lg (a^b) = b \lg a$:
$$= \frac{4 \lg 3}{\lg 7} \times \frac{2 \lg 10}{\lg 3} \times 2 \lg 7$$
- Simplify by canceling common terms:
$$= \frac{4 \cancel{\lg 3}}{\lg 7} \times \frac{2 \lg 10}{\cancel{\lg 3}} \times 2 \lg 7$$
- Multiply numerators and denominators:
$$= \frac{4 \times 2 \times 2 \times \lg 10 \times \lg 7}{\lg 7}$$
- Cancel $\lg 7$:
$$= 4 \times 2 \times 2 \times \lg 10 = 16 \times \lg 10$$
- Since $\lg 10 = 1$:
$$= 16$$
4. **Evaluate (f):**
- Use change of base for both logs with base 10:
$$\log_x 16 = \frac{\lg 16}{\lg x}, \quad \log_{64} x = \frac{\lg x}{\lg 64}$$
- Multiply:
$$\log_x 16 \times \log_{64} x = \frac{\lg 16}{\lg x} \times \frac{\lg x}{\lg 64}$$
- Cancel $\lg x$:
$$= \frac{\lg 16}{\lg 64}$$
- Express 16 and 64 as powers of 2:
$$16 = 2^4, \quad 64 = 2^6$$
- Substitute:
$$= \frac{\lg (2^4)}{\lg (2^6)} = \frac{4 \lg 2}{6 \lg 2}$$
- Cancel $\lg 2$:
$$= \frac{4}{6} = \frac{2}{3}$$
**Final answers:**
(a) $16$
(f) $\frac{2}{3}$