Exponential Equations Word Problems Worksheet - Free Printable

Let's solve each of these exponential equations without using logarithms, as the worksheet title suggests. We'll use properties of exponents, such as:

- $ a^m \cdot a^n = a^{m+n} $
- $ \frac{a^m}{a^n} = a^{m-n} $
- $ (a^m)^n = a^{mn} $
- $ a^0 = 1 $
- If $ a^x = a^y $, then $ x = y $ (for $ a > 0 $, $ a \ne 1 $)

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1) $ 4^{2x+3} = 1 $

We know that any non-zero number raised to the power 0 is 1.

So:
$$
4^{2x+3} = 1 = 4^0
\Rightarrow 2x + 3 = 0
\Rightarrow 2x = -3
\Rightarrow x = -\frac{3}{2}
$$

✔ Answer: $ x = -\frac{3}{2} $

---

2) $ 5^{3-2x} = 5^{-x} $

Same base, so set exponents equal:
$$
3 - 2x = -x
\Rightarrow 3 = x
$$

✔ Answer: $ x = 3 $

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3) $ 3^{1-2x} = 243 $

Note: $ 243 = 3^5 $, since $ 3^5 = 243 $

So:
$$
3^{1-2x} = 3^5
\Rightarrow 1 - 2x = 5
\Rightarrow -2x = 4
\Rightarrow x = -2
$$

✔ Answer: $ x = -2 $

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

Same base:
$$
2x = -x
\Rightarrow 3x = 0
\Rightarrow x = 0
$$

✔ Answer: $ x = 0 $

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5) $ 4^{3x-2} = 1 $

Again, anything to the 0 power is 1:
$$
4^{3x-2} = 4^0
\Rightarrow 3x - 2 = 0
\Rightarrow 3x = 2
\Rightarrow x = \frac{2}{3}
$$

✔ Answer: $ x = \frac{2}{3} $

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6) $ 4^{2x} = 4^{2x - 1} $

Same base:
$$
2x = 2x - 1
\Rightarrow 0 = -1
$$

This is a contradiction. No solution.

✔ Answer: No solution

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7) $ 6^{-2x} = 6^{1 - 3x} $

Same base:
$$
-2x = 1 - 3x
\Rightarrow -2x + 3x = 1
\Rightarrow x = 1
$$

✔ Answer: $ x = 1 $

---

8) $ 2^{2x+2} = 2^{3x} $

Same base:
$$
2x + 2 = 3x
\Rightarrow 2 = x
$$

✔ Answer: $ x = 2 $

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9) $ 6^{3m} \cdot 6^{-m} = 6^{-2m} $

Use product rule: $ 6^{3m + (-m)} = 6^{2m} $

So:
$$
6^{2m} = 6^{-2m}
\Rightarrow 2m = -2m
\Rightarrow 4m = 0
\Rightarrow m = 0
$$

✔ Answer: $ m = 0 $

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10) $ \frac{2^x}{2^1} = 2^{-2x} $

Simplify left side: $ 2^{x - 1} $

So:
$$
2^{x - 1} = 2^{-2x}
\Rightarrow x - 1 = -2x
\Rightarrow 3x = 1
\Rightarrow x = \frac{1}{3}
$$

✔ Answer: $ x = \frac{1}{3} $

---

11) $ 10^{-3x} \cdot 10^x = \frac{1}{10} $

Left side: $ 10^{-3x + x} = 10^{-2x} $

Right side: $ \frac{1}{10} = 10^{-1} $

So:
$$
10^{-2x} = 10^{-1}
\Rightarrow -2x = -1
\Rightarrow x = \frac{1}{2}
$$

✔ Answer: $ x = \frac{1}{2} $

---

12) $ 3^{-2x+1} \cdot 3^{-2x - 3} = 3^{-x} $

Add exponents on left:
$$
(-2x + 1) + (-2x - 3) = -4x - 2
$$

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

✔ Answer: $ x = -\frac{2}{3} $

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✔ Final Answers:

1) $ x = -\frac{3}{2} $
2) $ x = 3 $
3) $ x = -2 $
4) $ x = 0 $
5) $ x = \frac{2}{3} $
6) No solution
7) $ x = 1 $
8) $ x = 2 $
9) $ m = 0 $
10) $ x = \frac{1}{3} $
11) $ x = \frac{1}{2} $
12) $ x = -\frac{2}{3} $

Let me know if you'd like this formatted as a printable answer key!