Solve equations with variable exponents

Key Notes :

✨ Key Idea:

An exponential equation is an equation where the unknown variable is in the exponent.
Example: 2^x=8

We solve such equations by making bases the same or by using logarithms (at higher levels).

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🔹 Step 1: Rewrite with the Same Base

• If both sides can be written with the same base, equate the exponents.

Example 1:

2^x=8

Write 8 as 2³

2^x=2³

Since the bases are the same:

x=3

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🔹 Step 2: Use Properties of Exponents

Remember:

• a^m = aⁿ  ⟹  m = n (if a>0and a≠1)
• (a^m)ⁿ = a^m⋅n
• a^m⋅aⁿ = a^m+n

Example 2:

3^2x = 27

Write 27 as 3³

3^2x = 3³

2x = 3

x = 3/2

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🔹 Step 3: When Bases Are Different

• If bases cannot be rewritten to be the same, we use logarithms (optional for grade 8, but good to introduce).

Example 3:

5x = 12

Take log on both sides:

x⋅log⁡5 = log⁡12

x = log⁡12/log⁡5x

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🔹 Step 4: Equations with Exponential Expressions on Both Sides

Example 4:

4^x+1 = 16^x−2

4^x+1 = (4²)^x-2

4^x+1 = 4^2x-4

x+1 = 2x − 4

x = 5

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✅ Summary / Key Points

1. Rewrite numbers with the same base whenever possible.
2. If bases are the same → equate exponents.
3. Use laws of exponents to simplify.
4. If bases are different and cannot be matched → use logarithms.

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📝 Practice Problems

1. 2^x=32
2. 7^2x=343
3. 9^x+1=81^x−2
4. 3^x=20
5. 5^2x−1=25^x+2

Learn with an example

Let’s practice!🖊️