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Exponents – Running Notes
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Exponents (Level 1)

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Laws of Exponents Part 4

11. Equal Bases Law

If two powers have the same base and are equal, then their exponents are equal.

$a^m = a^n \Rightarrow m = n, \quad a \neq -1, 0, 1$

Example 1: If $2^{x+7} = 128$ , find $x$ .

We know:

$128 = 2^7$

So,

$2^{x+7} = 2^7$

Since the bases are equal,

$x + 7 = 7$

$x = 7 - 7 = 0$

Therefore,

$x = 0$

Example 2: If $5^{3x+9} = 625$ , find $x$ .

We know:

$625 = 5^4$

So,

$5^{3x+9} = 5^4$

Since the bases are equal,

$3x + 9 = 4$

$3x = 4 - 9$

$3x = -5$

$x = -\frac{5}{3}$

Example 3: If $4^{2x+7} = 64$ , find $x$ .

We know:

$64 = 4^3$

So,

$4^{2x+7} = 4^3$

Since the bases are equal,

$2x + 7 = 3$

$2x = 3 - 7$

$2x = -4$

$x = -2$

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