Power Laws of Logarithms

In logarithms, there are three fundamental power rules and here is the list of properties which represent formulas of power of the logarithms in algebraic form with proofs.

1.

Log of Exponential Term to a number

The logarithm of an exponential function to a number is equal to the product of the exponent of the exponential term and logarithm of base of the exponential term to the number.

$\large \log_{b} m^x = x \log_{b} m$

2.

Log of a number to Exponential Term

The logarithm of a number to an exponential function is equal to the product of reciprocal of the exponent of the base and logarithm of the number to base of the exponential term.

$\large \log_{b^y} m = \Big(\dfrac{1}{y}\Big) \log_{b} m$

3.

Log of Exponential term to another

The logarithm of an exponential function to another exponential term is equal to the product of the quotient of exponents of number by the base and logarithm of the base of the number to base of the base exponential term.

$\large \log_{b^y} m^x = \Big(\dfrac{x}{y}\Big) \log_{b} m$