IB Mathematics Analysis and Approaches Topic SL…

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

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

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

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

Consider a geometric sequence with $u_1=\dfrac{18}{5}$ , $r>0$ , and

u_2u_3u_4=\dfrac{729}{64}.

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

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

Show that $m = 2n - 1$ .

Consider the equation

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

where $x$ is real.

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

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

(c) 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

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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

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