Here is an example:

log₂(8³) = 3 ⋅ (log₂8)

log₂(8³) = 3 ⋅ 3

log₂(8³) = 9

Let's check this answer using the definition of logarithms. Since 8³ = 512 ,

log₂(8³) = log₂512

And since 2⁹ = 512

log₂512 = 9

Thus,

log₂(8³) = 9

In both cases, the result is 9.

The key to proving this theorem is the following exponentiation property:

x^m⋅n = (x^m)ⁿ

Quiz (1 point)

• x^{(m ⋅ n)} = (x^m)ⁿ

• if log_xz = p, then x^p = z

• if log_xz = p, then x^p = z

• if bⁿ = x, then (bⁿ)^p = x^p

• if the following are true:

  • a = x
  • b = y
  • x = y

  then a = b

• if b^m = bⁿ, then m = n

• if n ⋅ p = m, then m = p ⋅ n

• if the following are true:

  • a = b
  • b = c

  then a = c

• if log_bx = n, then p ⋅ (log_bx) = p ⋅ n

• if the following are true:

  • a = c
  • b = c

  then a = b

Please write your proof in the table below. Each row should contain one claim. The last claim is the statement that you are trying to prove.

Step	Claim	Reason (optional)	Error Message (if any)
1
2
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8
9
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