Exponentials & Logarithms

7 methods · 19 questions

★ Signature method unlocked: the technique this chapter rewards most often, with 3 worked questions.

The full chapter (7 methods · 19 worked questions) comes with Platform Access. See pricing

Whenever an equation is built from a single repeated exponential block (the same $a^{x}$ appearing as $a^{x}$ , $a^{2 x}$ , $a^{3 x}$ , a constant, ...), set $u$ equal to that block. Every term becomes a power of $u$ , the equation collapses to an ordinary polynomial (usually a quadratic), you solve for $u$ , then read off $x$ from $u = (base)^{x}$ .

Trigger: the variable appears only inside powers of one base, with exponents that are integer multiples of a single $x$ -term.

Instances:
(i) a plain $a^{2 x}$ and $a^{x}$ pattern gives a quadratic in $u = a^{x}$ ;
(ii) after rewriting to one base, $a^{2 x}$ becomes $u^{2}$ so even $4^{2 x}$ or $9^{x}$ fits;
(iii) a fractional exponent like $3^{x /2}$ is itself the right $u$ , with $3^{x} = u^{2}$ .

Linked questions (3)

TMUA 2016 Paper 1 · Q11

The real roots of the equation

4^{2 x} + 12 = 2^{2 x + 3}

are

p

and

q

, where

p > q

. The value of

p - q

can be expressed as

A $\frac{3}{4}$
B $1$
C $4$
D $- \frac{1}{2} + lo g_{10} \frac{3}{2}$
E $\frac{lo g _{10} 3}{lo g _{10} 4}$
F $\frac{lo g _{10} 3}{lo g _{10} 2}$

TMUA 2018 Paper 1 · Q15

Find the sum of the real solutions of the equation:

3^{x} - (3)^{x + 4} + 20 = 0

A $1$
B $4$
C $9$
D $lo g_{3} 20$
E $2 lo g_{3} 20$
F $4 lo g_{3} 20$

TMUA 2019 Paper 1 · Q15

Find the real non-zero solution to the equation

\frac{2 ^{(9^{x})}}{8 ^{(3^{x})}} = \frac{1}{4}

A $lo g_{3} 2$
B $2 lo g_{3} 2$
C $1$
D $2$
E $lo g_{2} 3$
F $2 lo g_{2} 3$