Simplify. Enter the result as a single logarithm with a coefficient of 1.
$\log _3(x)+\log _3\left(10 x^8\right)=$
Answer (1)
Apply the logarithm product rule: lo g b ( m ) + lo g b ( n ) = lo g b ( mn ) .
Combine the logarithms: lo g 3 ( x ) + lo g 3 ( 10 x 8 ) = lo g 3 ( x × 10 x 8 ) .
Simplify the expression inside the logarithm: x × 10 x 8 = 10 x 9 .
The simplified expression is lo g 3 ( 10 x 9 ) .
Explanation
Understanding the Problem We are given the expression lo g 3 ( x ) + lo g 3 ( 10 x 8 ) . Our goal is to simplify this expression into a single logarithm with a coefficient of 1.
Applying the Product Rule We will use the logarithm product rule, which states that lo g b ( m ) + lo g b ( n ) = lo g b ( mn ) . This rule allows us to combine the two logarithms into a single logarithm.
Combining the Logarithms Applying the product rule to the given expression, we have: lo g 3 ( x ) + lo g 3 ( 10 x 8 ) = lo g 3 ( x × 10 x 8 ) .
Simplifying the Expression Now, we simplify the expression inside the logarithm: x × 10 x 8 = 10 x 9 .
Final Answer Therefore, the simplified expression is: lo g 3 ( 10 x 9 ) . This is a single logarithm with a coefficient of 1.
Examples
Logarithms are used to solve exponential equations, which appear in various fields such as finance (calculating compound interest), physics (measuring radioactive decay), and computer science (analyzing algorithm complexity). For example, if you want to determine how long it will take for an investment to double at a certain interest rate, you would use logarithms to solve the exponential growth equation. Understanding logarithm properties allows for efficient manipulation and simplification of these equations.
