Product Rule

Reference > Algebra: Common Logarithms

Description

The Product Rule states that:

\(\log_{b}{(xy)}=\log_{b}{x}+\log_{b}{y}\)

Examples

Example 1

\[\log_{2}{9}+\log_{2}{3}\]

Use Product Rule: \(\log_{b}{(xy)}=\log_{b}{x}+\log_{b}{y}\).

\[\log_{2}{(9\times 3)}\]

Simplify \(9\times 3\) to \(27\).

\[\log_{2}{27}\]

Done

Decimal Form: 4.754888

Example 2

\[\log_{2}{2x}+\log_{2}{3y}\]

Use Product Rule: \(\log_{b}{(xy)}=\log_{b}{x}+\log_{b}{y}\).

\[\log_{2}{(2x\times 3y)}\]

Simplify \(2x\times 3y\) to \(6xy\).

\[\log_{2}{(6xy)}\]

Done

Example 3

\[\log_{2}{5}+\log_{2}{3}+\log_{2}{7}\]

Use Product Rule: \(\log_{b}{(xy)}=\log_{b}{x}+\log_{b}{y}\).

\[\log_{2}{(5\times 3\times 7)}\]

Simplify \(5\times 3\) to \(15\).

\[\log_{2}{(15\times 7)}\]

Simplify \(15\times 7\) to \(105\).

\[\log_{2}{105}\]

Done

Decimal Form: 6.714246