Exponential Equations

Q: How do I know when to use k-substitution?

When you see the same power, such as $2^x$, more than once, and one term is its square, such as $4^x = (2^x)^2$.

An exponential equation has the variable in the exponent. In Grade 11 you solve these by writing both sides with the same base or by factorising, without logarithms. These skills are needed for exponential graphs and finance.

Equations with a common base

If $a^{x} = a^{y}$ (with $a > 0$ , $a \neq = 1$ ), then $x = y$ .

Write both sides as powers of the same prime.
Equate the exponents.

Example. $2^{x - 1} = 32 = 2^{5}$ , so $x - 1 = 5$ and $x = 6$ .

Example. $9^{x} = 2 7^{x - 1}$

$3^{2 x} = 3^{3 x - 3} \Rightarrow 2 x = 3 x - 3 \Rightarrow x = 3$

Equations with rational exponents

If the variable is the base, raise both sides to the reciprocal power.

Example. $x^{\frac{3}{2}} = 8$

$x = 8^{\frac{2}{3}} = 4$

Even numerator. $x^{\frac{2}{3}} = 9$ means $(x^{\frac{1}{3}})^{2} = 9$ , so $x^{\frac{1}{3}} = \pm 3$ and $x = \pm 27$ . Both work, because cube roots of negative numbers exist.

Check answers in the original equation.

Equations that factorise into exponential terms

Common factor. $5^{x + 1} - 5^{x} = 100$

$5^{x} (5 - 1) = 100 \Rightarrow 5^{x} = 25 \Rightarrow x = 2$

Quadratic form. Let $k = 3^{x}$ in $3^{2 x} - 10 \cdot 3^{x} + 9 = 0$ :

$k^{2} - 10 k + 9 = 0 \Rightarrow (k - 1) (k - 9) = 0$

$3^{x} = 1$ gives $x = 0$ , and $3^{x} = 9$ gives $x = 2$ .

Reject any value where $3^{x}$ would be zero or negative, because $a^{x} > 0$ always.

Key terms

Exponential equation: An equation with the variable in an exponent.
Common base: Writing both sides as powers of the same number.
k-substitution: Replacing a repeated expression like $2^{x}$ with $k$ .

Common mistakes

Equating exponents before the bases are the same.
Writing $5^{x + 1} - 5^{x} = 5^{1}$ .
Keeping a solution like $2^{x} = - 4$ .
Missing the negative root when solving $x^{\frac{2}{3}} = 9$ .

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

3 marks total

knowledge

Solve for $x$ : $2^{x - 1} = 32$

Worked solution

Hidden - try the question first

Solution

$32 = 2^{5}$
$x - 1 = 5$
$x = 6$

Question 2

5 marks total

routine

Solve for $x$ :

(a) $9^{x} = 2 7^{x - 1}$ (3)

(b) $x^{\frac{3}{2}} = 8$ (2)

Worked solution

Hidden - try the question first

Solution

(a) $3^{2 x} = 3^{3 x - 3}$
$2 x = 3 x - 3$
$x = 3$
(b) $x = 8^{\frac{2}{3}}$
$= 4$

Question 3

6 marks total

complex

Solve for $x$ :

(a) $5^{x + 1} - 5^{x} = 100$ (3)

(b) $3^{2 x} - 10 \cdot 3^{x} + 9 = 0$ (3)

Worked solution

Hidden - try the question first

Solution

(a) $3^{2 x} = 3^{3 x - 3}$
$5^{x} = 25$
$x = 2$
(b) $(3^{x} - 1) (3^{x} - 9) = 0$
$3^{x} = 1 \Rightarrow x = 0$
$3^{x} = 9 \Rightarrow x = 2$

Question 4

8 marks total

problem-solving

Solve for $x$ :

(a) $2 \cdot 4^{x} - 9 \cdot 2^{x} + 4 = 0$ (5)

(b) $x^{\frac{2}{3}} = 9$ (3)

Worked solution

Hidden - try the question first

Solution

(a) $3^{2 x} = 3^{3 x - 3}$ ; let $k = 2^{x}$
$x = 3$
$(2 k - 1) (k - 4) = 0$
$2^{x} = \frac{1}{2} \Rightarrow x = - 1$
$2^{x} = 4 \Rightarrow x = 2$
(b) $(x^{\frac{1}{3}})^{2} = 9 \Rightarrow x^{\frac{1}{3}} = \pm 3$
$x = 27$
or $x = - 27$

FAQs

Do I use logarithms in Grade 11?

No. Grade 11 exponential equations are solved by matching bases or factorising. Logarithms are Grade 12 work.

How do I know when to use k-substitution?

When you see the same power, such as $2^{x}$ , more than once, and one term is its square, such as $4^{x} = (2^{x})^{2}$ .

Can 2 to the power x equal a negative number?

No. A positive base to any power is positive, so reject those solutions.

Equations with a common base

Equations with rational exponents

Equations that factorise into exponential terms

Key terms

Common mistakes

Practice questions

FAQs

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