Laws of Logarithms - IB DP Mathematics AA SL - Past Papers

A sequence starts with 4 and increases in such a way that the second term is 8 and the third term is 12.
(a) Determine the constant increment, $c$, for the sequence.
(b) Calculate the following:
(i) The 15th term of the sequence, $t_{15}$;
(ii) The sum of the first 15 terms, $S_{15}$.
(c) If the term $t_m = 364$, find the value of $m$.

(a) Calculate the value of:
(i) $\log_3 27$
(ii) $\log_4 \left( \frac{1}{16} \right)$
(iii) $\log_{10} 1000$

(b) Hence, solve $\log_3 27 + \log_4 \left( \frac{1}{16} \right) + \log_{10} 1000 = \log_2 y$.

Solve $3\ln 4 - \ln 16 = -\ln y$ for $y$.

Rewrite the expression $4\ln 3 - \ln 9$ in the form $\ln k$, where $k \in \mathbb{Z}$.

Let $a = \ln 4$ and $b = \ln 8$. Write down the following expressions in terms of $a$ and $b$:
a. $\ln 16$
b. $\ln 2$
c. $\ln 64$

Determine the value of each of the following.

a.

$\log_{5} 5$

b.

$\log_{5} 25 + \log_{5} 2$

c.

$\log_{5} 100 - \log_{5} 4$