Exponential equations questions with answers

Questions

1. Question 1Same base

   Solve $2^{2x - 3} = 8$.

   Show the worked solution

   1. Step 1: Write 8 as a power of 2$8 = 2^{3}$
   2. Step 2: Write both sides with the same base$2^{2x - 3} = 2^{3}$
   3. Step 3: Same base: the powers are equal$2x - 3 = 3$$2x = 6$$x = 3$
2. Question 2Same base

   Solve $5^{x - 2} = 25$.

   Show the worked solution

   1. Step 1: Write 25 as a power of 5$25 = 5^{2}$
   2. Step 2: Write both sides with the same base$5^{x - 2} = 5^{2}$
   3. Step 3: Same base: the powers are equal$x - 2 = 2$$x = 4$
3. Question 3Make the bases the same

   Solve $9^{x} = 3^{x - 1}$.

   Show the worked solution

   1. Step 1: Write both bases as powers of the same number$9 = 3^{2}$$3 = 3^{1}$
   2. Step 2: Rewrite the equation with base 3, then multiply the powers: (aᵐ)ⁿ = aᵐⁿ$(3^{2})^{x}$ $=$ $(3^{1})^{x - 1}$$3^{2x} = 3^{(x - 1)}$
   3. Step 3: Same base: the powers are equal. Expand and solve$2x = (x - 1)$$2x$ $=$ $x - 1$$x = -1$
4. Question 4Make the bases the same

   Solve $3^{x - 1} = 9^{2x}$.

   Show the worked solution

   1. Step 1: Write both bases as powers of the same number$3 = 3^{1}$$9 = 3^{2}$
   2. Step 2: Rewrite the equation with base 3, then multiply the powers: (aᵐ)ⁿ = aᵐⁿ$(3^{1})^{x - 1}$ $=$ $(3^{2})^{2x}$$3^{(x - 1)} = 3^{2(2x)}$
   3. Step 3: Same base: the powers are equal. Expand and solve$(x - 1) = 2(2x)$$x - 1$ $=$ $4x$$-3x = 1$$x = -\dfrac{1}{3}$
5. Question 5Hidden quadratic

   Solve $5^{2x} - 0(5^x) - 25 = 0$.

   Show the worked solution

   1. Step 1: Write 5²ˣ as (5ˣ)²: then it is a quadratic in 5ˣ$5^{2x} = (5^{x})^{2}$$(5^{x})^2 - 25 = 0$
   2. Step 2: Let y = 5ˣ$y^2 - 25 = 0$
   3. Step 3: Factorise and solve for y$(y + 5)(y - 5) = 0$$y = -5$ or $y = 5$
   4. Step 4: Write each y as a power of 5 (5ˣ is always positive)$5^x = -5$ has no solution, since $5^x > 0$.$5^x = 5$$5^x = 5^{1}$$x = 1$
6. Question 6Hidden quadratic

   Solve $3^{2x} - 10(3^x) + 9 = 0$.

   Show the worked solution

   1. Step 1: Write 3²ˣ as (3ˣ)²: then it is a quadratic in 3ˣ$3^{2x} = (3^{x})^{2}$$(3^{x})^2 - 10(3^{x}) + 9 = 0$
   2. Step 2: Let y = 3ˣ$y^2 - 10y + 9 = 0$
   3. Step 3: Factorise and solve for y$(y - 1)(y - 9) = 0$$y = 1$ or $y = 9$
   4. Step 4: Write each y as a power of 3 (3ˣ is always positive)$3^x = 1$$3^x = 3^{0}$$x = 0$$3^x = 9$$3^x = 3^{2}$$x = 2$