Exponential equations practice worksheet with problems involving powers of 3, 5, 8, 4, 2, and other bases.

Let's solve each of these exponential equations step by step. We'll use properties of exponents and logarithms where necessary, but most can be solved by expressing both sides with the same base.

---

a) $ 3^{2x} = 9 $

We know $ 9 = 3^2 $, so:

$$
3^{2x} = 3^2 \Rightarrow 2x = 2 \Rightarrow x = 1
$$

✔ Answer: $ x = 1 $

---

b) $ 5^{3x+2} = 25^{x-1} $

Note $ 25 = 5^2 $, so:

$$
5^{3x+2} = (5^2)^{x-1} = 5^{2(x-1)} = 5^{2x - 2}
$$

Now equate exponents:

$$
3x + 2 = 2x - 2 \Rightarrow x = -4
$$

✔ Answer: $ x = -4 $

---

c) $ 2^{3-x} = 4^{2-x} $

$ 4 = 2^2 $, so:

$$
2^{3-x} = (2^2)^{2-x} = 2^{2(2-x)} = 2^{4 - 2x}
$$

Equate exponents:

$$
3 - x = 4 - 2x \Rightarrow x = 1
$$

✔ Answer: $ x = 1 $

---

d) $ 8^{-x} = 2^{x-8} $

$ 8 = 2^3 $, so:

$$
(2^3)^{-x} = 2^{-3x}, \quad \text{so: } 2^{-3x} = 2^{x - 8}
$$

Equate exponents:

$$
-3x = x - 8 \Rightarrow -4x = -8 \Rightarrow x = 2
$$

✔ Answer: $ x = 2 $

---

e) $ \left(\frac{1}{4}\right)^{x-1} = 4^{2-3x} $

Note $ \frac{1}{4} = 4^{-1} $, so:

$$
(4^{-1})^{x-1} = 4^{-(x-1)} = 4^{-x+1}
$$

Right side: $ 4^{2 - 3x} $

So:

$$
4^{-x+1} = 4^{2 - 3x} \Rightarrow -x + 1 = 2 - 3x \Rightarrow 2x = 1 \Rightarrow x = \frac{1}{2}
$$

✔ Answer: $ x = \frac{1}{2} $

---

f) $ 5^{3x+2} = 25^{x-1} $

Same as (b): $ 25 = 5^2 $

$$
5^{3x+2} = (5^2)^{x-1} = 5^{2x - 2}
\Rightarrow 3x + 2 = 2x - 2 \Rightarrow x = -4
$$

✔ Answer: $ x = -4 $

---

g) $ 4^x = 2^{x+1} $

$ 4 = 2^2 $, so:

$$
(2^2)^x = 2^{2x}, \quad \text{so: } 2^{2x} = 2^{x+1}
\Rightarrow 2x = x + 1 \Rightarrow x = 1
$$

✔ Answer: $ x = 1 $

---

h) $ 8^{2x+1} = \left(\frac{1}{16}\right)^{3-2x} $

First: $ 8 = 2^3 $, $ 16 = 2^4 $, so $ \frac{1}{16} = 2^{-4} $

Left: $ (2^3)^{2x+1} = 2^{3(2x+1)} = 2^{6x + 3} $

Right: $ (2^{-4})^{3-2x} = 2^{-4(3-2x)} = 2^{-12 + 8x} $

Set equal:

$$
6x + 3 = -12 + 8x \Rightarrow 3 + 12 = 8x - 6x \Rightarrow 15 = 2x \Rightarrow x = \frac{15}{2}
$$

✔ Answer: $ x = \frac{15}{2} $

---

i) $ \left(\frac{1}{2}\right)^{3x} = 2^{-x-1} $

Note $ \frac{1}{2} = 2^{-1} $, so:

$$
(2^{-1})^{3x} = 2^{-3x}
\Rightarrow 2^{-3x} = 2^{-x - 1}
\Rightarrow -3x = -x - 1 \Rightarrow -2x = -1 \Rightarrow x = \frac{1}{2}
$$

✔ Answer: $ x = \frac{1}{2} $

---

j) $ 2^{-3x+1} = \left(\frac{1}{2}\right)^{-x-1} $

Note $ \frac{1}{2} = 2^{-1} $, so right side:

$$
(2^{-1})^{-x-1} = 2^{x+1}
$$

Left side: $ 2^{-3x+1} $

So:

$$
-3x + 1 = x + 1 \Rightarrow -4x = 0 \Rightarrow x = 0
$$

✔ Answer: $ x = 0 $

---

k) $ 2^{x-1} = \left(\frac{1}{4}\right)^{2-4x} $

$ \frac{1}{4} = 2^{-2} $, so:

Right: $ (2^{-2})^{2-4x} = 2^{-2(2-4x)} = 2^{-4 + 8x} $

Left: $ 2^{x - 1} $

So:

$$
x - 1 = -4 + 8x \Rightarrow -1 + 4 = 8x - x \Rightarrow 3 = 7x \Rightarrow x = \frac{3}{7}
$$

✔ Answer: $ x = \frac{3}{7} $

---

l) $ \left(\frac{1}{27}\right)^{4-x} = 9^{2x} $

$ \frac{1}{27} = 3^{-3} $, $ 9 = 3^2 $

Left: $ (3^{-3})^{4-x} = 3^{-3(4-x)} = 3^{-12 + 3x} $

Right: $ (3^2)^{2x} = 3^{4x} $

So:

$$
-12 + 3x = 4x \Rightarrow -12 = x \Rightarrow x = -12
$$

✔ Answer: $ x = -12 $

---

m) $ \left(\frac{3}{5}\right)^x = \left(\frac{5}{3}\right)^3 $

Note $ \frac{5}{3} = \left(\frac{3}{5}\right)^{-1} $, so:

Right: $ \left(\frac{3}{5}\right)^{-3} $

So:

$$
\left(\frac{3}{5}\right)^x = \left(\frac{3}{5}\right)^{-3} \Rightarrow x = -3
$$

✔ Answer: $ x = -3 $

---

n) $ 2^{3x-4} = \left(\frac{1}{8}\right)^{x+1} $

$ \frac{1}{8} = 2^{-3} $, so:

Right: $ (2^{-3})^{x+1} = 2^{-3(x+1)} = 2^{-3x - 3} $

Left: $ 2^{3x - 4} $

So:

$$
3x - 4 = -3x - 3 \Rightarrow 6x = 1 \Rightarrow x = \frac{1}{6}
$$

✔ Answer: $ x = \frac{1}{6} $

---

o) $ 4^{1-x} = \left(\frac{1}{4}\right)^{2x-3} $

$ 4 = 2^2 $, $ \frac{1}{4} = 4^{-1} $, so:

Left: $ (4)^{1-x} = 4^{1-x} $

Right: $ (4^{-1})^{2x-3} = 4^{-(2x-3)} = 4^{-2x + 3} $

So:

$$
1 - x = -2x + 3 \Rightarrow x = 2
$$

✔ Answer: $ x = 2 $

---

p) $ 10^x = 0.1 \times 1000^{x-1} $

Write all in powers of 10:

- $ 0.1 = 10^{-1} $
- $ 1000 = 10^3 $, so $ 1000^{x-1} = (10^3)^{x-1} = 10^{3(x-1)} $

So:

$$
10^x = 10^{-1} \cdot 10^{3x - 3} = 10^{3x - 4}
\Rightarrow x = 3x - 4 \Rightarrow -2x = -4 \Rightarrow x = 2
$$

✔ Answer: $ x = 2 $

---

q) $ 27 \times 27^{x-3} = 81^{3x-5} $

Note:
- $ 27 = 3^3 $
- $ 81 = 3^4 $

Left: $ 27^1 \cdot 27^{x-3} = 27^{x-2} = (3^3)^{x-2} = 3^{3(x-2)} = 3^{3x - 6} $

Right: $ (3^4)^{3x-5} = 3^{4(3x-5)} = 3^{12x - 20} $

So:

$$
3x - 6 = 12x - 20 \Rightarrow -9x = -14 \Rightarrow x = \frac{14}{9}
$$

✔ Answer: $ x = \frac{14}{9} $

---

r) $ 4 \times 2^{x+1} = \left(\frac{1}{8}\right)^{2x-3} $

Left: $ 4 = 2^2 $, so:

$$
2^2 \cdot 2^{x+1} = 2^{x+3}
$$

Right: $ \frac{1}{8} = 2^{-3} $, so:

$$
(2^{-3})^{2x-3} = 2^{-3(2x-3)} = 2^{-6x + 9}
$$

So:

$$
x + 3 = -6x + 9 \Rightarrow 7x = 6 \Rightarrow x = \frac{6}{7}
$$

✔ Answer: $ x = \frac{6}{7} $

---

s) $ 4^x = 16^{-x} $

$ 4 = 2^2 $, $ 16 = 2^4 $, so:

Left: $ (2^2)^x = 2^{2x} $

Right: $ (2^4)^{-x} = 2^{-4x} $

So:

$$
2x = -4x \Rightarrow 6x = 0 \Rightarrow x = 0
$$

✔ Answer: $ x = 0 $

---

t) $ 4^{x-2} = 0.125 $

$ 0.125 = \frac{1}{8} = 2^{-3} $

$ 4 = 2^2 $, so:

$$
(2^2)^{x-2} = 2^{2(x-2)} = 2^{2x - 4}
\Rightarrow 2^{2x - 4} = 2^{-3}
\Rightarrow 2x - 4 = -3 \Rightarrow 2x = 1 \Rightarrow x = \frac{1}{2}
$$

✔ Answer: $ x = \frac{1}{2} $

---

u) $ \left(\frac{1}{8}\right)^{-x} = \left(\frac{1}{32}\right)^{1-x} $

$ \frac{1}{8} = 2^{-3} $, $ \frac{1}{32} = 2^{-5} $

Left: $ (2^{-3})^{-x} = 2^{3x} $

Right: $ (2^{-5})^{1-x} = 2^{-5(1-x)} = 2^{-5 + 5x} $

So:

$$
3x = -5 + 5x \Rightarrow -2x = -5 \Rightarrow x = \frac{5}{2}
$$

✔ Answer: $ x = \frac{5}{2} $

---

v) $ \left(\frac{1}{243}\right)^{2x} = 81^{1-x} $

$ 243 = 3^5 $, so $ \frac{1}{243} = 3^{-5} $

Left: $ (3^{-5})^{2x} = 3^{-10x} $

$ 81 = 3^4 $, so right: $ (3^4)^{1-x} = 3^{4(1-x)} = 3^{4 - 4x} $

So:

$$
-10x = 4 - 4x \Rightarrow -6x = 4 \Rightarrow x = -\frac{2}{3}
$$

✔ Answer: $ x = -\frac{2}{3} $

---

w) $ \left(\frac{1}{125}\right)^{-3x+1} = 25^{-x-1} $

$ 125 = 5^3 $, so $ \frac{1}{125} = 5^{-3} $

Left: $ (5^{-3})^{-3x+1} = 5^{(-3)(-3x+1)} = 5^{9x - 3} $

$ 25 = 5^2 $, so right: $ (5^2)^{-x-1} = 5^{-2x - 2} $

So:

$$
9x - 3 = -2x - 2 \Rightarrow 11x = 1 \Rightarrow x = \frac{1}{11}
$$

✔ Answer: $ x = \frac{1}{11} $

---

x) $ 3^{x-2} = \left(\frac{1}{3}\right)^{-2x} $

$ \frac{1}{3} = 3^{-1} $, so right:

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

Left: $ 3^{x-2} $

So:

$$
x - 2 = 2x \Rightarrow -2 = x \Rightarrow x = -2
$$

✔ Answer: $ x = -2 $

---

y) $ \left(\frac{1}{3}\right)^{1-x} = 243^{-x-1} $

$ \frac{1}{3} = 3^{-1} $, so left: $ (3^{-1})^{1-x} = 3^{-(1-x)} = 3^{x-1} $

$ 243 = 3^5 $, so right: $ (3^5)^{-x-1} = 3^{-5x - 5} $

So:

$$
x - 1 = -5x - 5 \Rightarrow 6x = -4 \Rightarrow x = -\frac{2}{3}
$$

✔ Answer: $ x = -\frac{2}{3} $

---

z) $ 5^{3x+2-(1-x)} = 125^{2x-1-x} $

Simplify exponents:

Left exponent: $ 3x + 2 - (1 - x) = 3x + 2 - 1 + x = 4x + 1 $

So left: $ 5^{4x+1} $

Right: $ 125^{2x - 1 - x} = 125^{x - 1} $

$ 125 = 5^3 $, so:

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

So:

$$
4x + 1 = 3x - 3 \Rightarrow x = -4
$$

✔ Answer: $ x = -4 $

---

## ✔ Final Answers Summary:

| Letter | Answer |
|--------|---------------|
| a) | $ x = 1 $ |
| b) | $ x = -4 $ |
| c) | $ x = 1 $ |
| d) | $ x = 2 $ |
| e) | $ x = \frac{1}{2} $ |
| f) | $ x = -4 $ |
| g) | $ x = 1 $ |
| h) | $ x = \frac{15}{2} $ |
| i) | $ x = \frac{1}{2} $ |
| j) | $ x = 0 $ |
| k) | $ x = \frac{3}{7} $ |
| l) | $ x = -12 $ |
| m) | $ x = -3 $ |
| n) | $ x = \frac{1}{6} $ |
| o) | $ x = 2 $ |
| p) | $ x = 2 $ |
| q) | $ x = \frac{14}{9} $ |
| r) | $ x = \frac{6}{7} $ |
| s) | $ x = 0 $ |
| t) | $ x = \frac{1}{2} $ |
| u) | $ x = \frac{5}{2} $ |
| v) | $ x = -\frac{2}{3} $ |
| w) | $ x = \frac{1}{11} $ |
| x) | $ x = -2 $ |
| y) | $ x = -\frac{2}{3} $ |
| z) | $ x = -4 $ |

Let me know if you'd like any explanation for a specific one!