Question

4.
a) $\log1000 = $

b) $\log_{16}8 = $

c) $\log_{18}1 = $

d) $\log_{5}38 = $

Explanation:

Step1: Evaluate common log of 1000

Recall $\log_{10}10^n = n$. Since $1000=10^3$, $\log 1000 = \log_{10}10^3 = 3$

Step2: Evaluate base-16 log of 8

Rewrite 16 and 8 as powers of 2: $16=2^4$, $8=2^3$. Use log rule $\log_{b^k}a^m=\frac{m}{k}\log_b a$.
$\log_{16}8 = \log_{2^4}2^3 = \frac{3}{4}\log_2 2 = \frac{3}{4}$

Step3: Evaluate log of 1 (base 18)

Use log rule $\log_b 1 = 0$ for any $b>0, b
eq1$. So $\log_{18}1 = 0$

Step4: Evaluate base-5 log of 38

Use change of base formula $\log_b a = \frac{\ln a}{\ln b}$.
$\log_5 38 = \frac{\ln 38}{\ln 5} \approx \frac{3.6376}{1.6094} \approx 2.260$

Answer:

A) $3$
B) $\frac{3}{4}$
C) $0$
D) $\approx 2.26$