IB Mathematics Analysis and Approaches Topic SL…

Consider the infinite series

S = \log_a(1) + \log_a(r) + \log_a(r^2) + \log_a(r^3) + \dots ,

where $a>1$ and $0<r<1$ .

Show that the $n$ -th partial sum can be written as

S_n = \log_a(r^{0+1+2+\dots+(n-1)}).

Let $a > 1$ and $b > 1$ .

Consider a geometric sequence with $u_1=\dfrac{18}{5}$ , , and

Given that $\log_5 K = m$ and $\log_{25} (5$ ,

Consider the equation

$\left(x^{2}-5x+7\right)^{x^{2}-4x+3} = 1,$

that

S_n=\frac{n(n-1)}{2}\,\log_a(r).

Hence, why the infinite series diverges to $-\infty$ .

Show that $\log_{a} b + \log_{b} a \ge 2$ , and find the condition on $a$ and $b$ under which equality holds.

Given that $\log_{a} b + \log_{b} a = \frac{5}{2}$ , show that $b = a^{2}$ or $b^{2} = a$ .

Hence find all pairs of integers $(a,\,b)$ with $a > 1$ , $b > 1$ , $\log_{a} b + \log_{b} a = \frac{5}{2}$ and $a + b \le 20$ .

Find the least value of $n$ such that $u_n < 1$ .

Show that $m = 2n - 1$ .

Find the values of $x$ for which the exponent is zero, and verify that the equation holds for each of them.

Find the values of $x$ for which the base is equal to $1$ .

Show that $x^{2}-5x+7 > 0$ for every real $x$ , and hence write down the complete solution set of the equation, showing that it has exactly three elements.

IB Mathematics Analysis and Approaches SL 1.7—laws of Exponents and Logs Questionbank

IB Mathematics Analysis and Approaches SL 1.7—laws of Exponents and Logs Questionbank

IB Mathematics Analysis and Approaches SL 1.7—laws of Exponents and Logs Questionbank

IB Mathematics Analysis and Approaches SL 1.7—laws of Exponents and Logs Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 1.7—laws of Exponents and Logs Questionbank