Properties of Logarithms

The properties of logarithms enable us to change the form of logarithmic statements so that products can be converted to sums, quotients can be converted to differences, and powers can be converted to products.

Properties of Logarithms

Product Property

The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.

For $LaTeX: x>0,\:y>0,\:a>0,\:a\ne1,\:$ $x>0,\:y>0,\:a>0,\:a\ne1,\:$ and any real number $LaTeX: r,\:$ $r,\:$ the following property holds.

$LaTeX: \log_axy=\log_ax+\log_ay$ $\log_axy=\log_ax+\log_ay$

Quotient Property

The logarithm of the quotient of two numbers is equal to the difference between the logarithms of the numbers.

For $LaTeX: x>0,\:y>0,\:a>0,\:a\ne1,\:$ $x>0,\:y>0,\:a>0,\:a\ne1,\:$ and any real number $LaTeX: r,\:$ $r,\:$ the following property holds.

$LaTeX: \log_a\frac{x}{y}=\log_ax-\log_ay$ $\log_a\frac{x}{y}=\log_ax-\log_ay$

Power Property

The logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the number.

For $LaTeX: x>0,\:y>0,\:a>0,\:a\ne1,\:$ $x>0,\:y>0,\:a>0,\:a\ne1,\:$ and any real number $LaTeX: r,\:$ $r,\:$ the following property holds.

$LaTeX: \log_ax^r=rlog_ax$ $\log_ax^r=rlog_ax$

Logarithm of 1

The base LaTeX: a $a$ logarithm of 1 is 0.

For $LaTeX: x>0,\:y>0,\:a>0,\:a\ne1,\:$ $x>0,\:y>0,\:a>0,\:a\ne1,\:$ and any real number $LaTeX: r,\:$ $r,\:$ the following property holds.

$LaTeX: \log_a1=0$ $\log_a1=0$

Base a Logarithm

The base LaTeX: a $a$ logarithm of $a$ is 1.

For $LaTeX: x>0,\:y>0,\:a>0,\:a\ne1,\:$ $x>0,\:y>0,\:a>0,\:a\ne1,\:$ and any real number $LaTeX: r,\:$ $r,\:$ the following property holds.

$LaTeX: \log_aa=1$ $\log_aa=1$