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Logarithm Power Rule

The logarithm power rule is:

log b(x) r = r · log b(x)    b,x ∈ R +, r ∈ R, b ≠ 1

R + denotes the set of positive real numbers.

Logarithm Power Rule Proof

x r = (b (log b(x) ) r = b r · logb (x)

⇒ log bx r = r · log b(x)

Because, by definition of the logarithm:

b y = z ⇔ y = log b(z).

Logarithm Power Rule Examples

log 10(10) 2 = 2 · log 10(10) = 2 · 1 = 2 log 10(10000) = log (10) 4 = 4 · log (10) 4 = 4 · 1 = 4

Binary Log Power Rule Examples

log 2(x) r = r · log 2(x)    b,x ∈ R +, r ∈ R, b ≠ 1

log 2(2) 3 = 3 · log 2(2) = 3 · 1 = 3 log 2(64) = log 2(2) 6 = 6 · log 2(2) = 6 · 1 = 6

Natural Log Power Rule Examples

ln(x) r = r · ln(x)    b,x ∈ R +, r ∈ R, b ≠ 1

ln(e) 2 = 2 · ln(e) = 2 · 1 = 2 ln(25) = ln(5) 2 = 2 · ln(5) = 2 · 1.6094379124341005 = 3.218875824868201

Frequently Asked Questions

Click on the question which is of interest to you to see the collapsible content answer.
The value of any log with a given argument in exponential form is equal to the exponent multiplied by the logarithm of the argument.
log b(x) r = r · log b(x)
Yes, because y = log b(x) = y ⇔ b y
The coefficient r in r · log b(x) is equal to the exponent of log b(x) r.
The inverse of a logarithmic function is an exponential function – and vice versa. In other words, the two functions undo each other.

Summary


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