Simplify. Enter the result as a single logarithm with a coefficient of 1.
$\log _8\left(3 x^6\right)+\log _8\left(2 x^3\right)=$
Answer (1)
Use the logarithm property lo g b ( m ) + lo g b ( n ) = lo g b ( mn ) to combine the logarithms.
Multiply the expressions inside the logarithms: ( 3 x 6 ) ( 2 x 3 ) = 6 x 9 .
Express the result as a single logarithm: lo g 8 ( 6 x 9 ) .
The simplified expression is lo g 8 ( 6 x 9 ) .
Explanation
Understanding the problem We are given the expression lo g 8 ( 3 x 6 ) + lo g 8 ( 2 x 3 ) . We need to simplify this expression into a single logarithm with a coefficient of 1.
Applying the Logarithm Property We will use the property of logarithms that states lo g b ( m ) + lo g b ( n ) = lo g b ( mn ) . This property allows us to combine the two logarithms into a single logarithm.
Combining the Logarithms Applying the property, we have: lo g 8 ( 3 x 6 ) + lo g 8 ( 2 x 3 ) = lo g 8 (( 3 x 6 ) ( 2 x 3 ))
Simplifying the Expression Now, we simplify the expression inside the logarithm: ( 3 x 6 ) ( 2 x 3 ) = 3 × 2 × x 6 × x 3 = 6 x 6 + 3 = 6 x 9
Final Simplified Expression Therefore, the simplified expression is: lo g 8 ( 6 x 9 )
Examples
Logarithms are used in many scientific and engineering fields. For example, the Richter scale, which measures the magnitude of earthquakes, is a logarithmic scale. Similarly, the pH scale, which measures the acidity or alkalinity of a solution, is also a logarithmic scale. Understanding how to simplify logarithmic expressions can help in analyzing and interpreting data in these fields.
