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# Logarithm Power Rule

The logarithm power rule is:

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

R+ denotes the set of positive real numbers.

## Logarithm Power Rule Proof

xr = (b(logb(x))r = b r · logb (x)

logbxr = r · log b(x)

Because, by definition of the logarithm:

by = z ⇔ y = logb(z).

## Logarithm Power Rule Examples

log10(10)2 = 2 · log 10(10) = 2 · 1 = 2 log10(10000) = log (10)4 = 4 · log (10)4 = 4 · 1 = 4

### Binary Log Power Rule Examples

log2(x)r = r · log2(x)    b,x ∈ R+, r ∈ R, b ≠ 1

log2(2)3 = 3 · log 2(2) = 3 · 1 = 3 log2(64) = log2(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

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### How do you solve logarithmic powers?

The value of any log with a given argument in exponential form is equal to the exponent multiplied by the logarithm of the argument.

### What is the power rule for logarithms?

logb(x)r = r · logb(x)

### Can logarithms be exponents?

Yes, because y = log b(x) = y ⇔ b y

### What does a number before a log mean?

The coefficient r in r · logb(x) is equal to the exponent of logb(x)r.

### How do you convert power to log?

The inverse of a logarithmic function is an exponential function – and vice versa. In other words, the two functions undo each other.

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