Power rule of Logarithms

`log_a X^p = plog_a X`

Subjects > Mathematics > Logarithms

The power rule of Logarithms

Provided that `X > 0`, and `a ne 1` and `m, p` is any real number, then
`log_a X^p = plog_a X`

Proof

let `X = a^m` which gives `log_a X = m`

From, `log_a (X^p) = log_a (X)^p` - Law of exponents

⇒ `log_a (X^p) = log_a (a^m)^p` - By substituting the value of `X`

⇒ `log_a (X^p) = log_a (a^(mp))` - Law of exponents

⇒ `log_a X^p = mp`

⇒ `log_a X^p = plog_a X` - By substituting the value of `m`
∴ Hence proved.

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The power rule of Logarithms

Provided that `X > 0`, and `a ne 1` and `m, p` is any real number, then
`log_a X^p = plog_a X`

Proof

let `X = a^m` which gives `log_a X = m`

From, `log_a (X^p) = log_a (X)^p` - Law of exponents

⇒ `log_a (X^p) = log_a (a^m)^p` - By substituting the value of `X`

⇒ `log_a (X^p) = log_a (a^(mp))` - Law of exponents

⇒ `log_a X^p = mp`

⇒ `log_a X^p = plog_a X` - By substituting the value of `m`
∴ Hence proved.

Recommended Lessons