Product Rule

Reference > Algebra: Common Logarithms

Description

The Product Rule states that:

log⁡b(xy)=log⁡bx+log⁡by\log_{b}{(xy)}=\log_{b}{x}+\log_{b}{y}
Examples

Example 1

log⁡29+log⁡23\log_{2}{9}+\log_{2}{3}
1
Use Product Rule: log⁡b(xy)=log⁡bx+log⁡by\log_{b}{(xy)}=\log_{b}{x}+\log_{b}{y}.
log⁡2(9×3)\log_{2}{(9\times 3)}

2
Simplify  9×39\times 3  to  2727.
log⁡227\log_{2}{27}

Done

Decimal Form: 4.754888


 

Example 2

log⁡22x+log⁡23y\log_{2}{2x}+\log_{2}{3y}
1
Use Product Rule: log⁡b(xy)=log⁡bx+log⁡by\log_{b}{(xy)}=\log_{b}{x}+\log_{b}{y}.
log⁡2(2x×3y)\log_{2}{(2x\times 3y)}

2
Simplify  2x×3y2x\times 3y  to  6xy6xy.
log⁡2(6xy)\log_{2}{(6xy)}

Done


 

Example 3

log⁡25+log⁡23+log⁡27\log_{2}{5}+\log_{2}{3}+\log_{2}{7}
1
Use Product Rule: log⁡b(xy)=log⁡bx+log⁡by\log_{b}{(xy)}=\log_{b}{x}+\log_{b}{y}.
log⁡2(5×3×7)\log_{2}{(5\times 3\times 7)}

2
Simplify  5×35\times 3  to  1515.
log⁡2(15×7)\log_{2}{(15\times 7)}

3
Simplify  15×715\times 7  to  105105.
log⁡2105\log_{2}{105}

Done

Decimal Form: 6.714246


 
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