What is $\log _5(4 \cdot 7)+\log _5 2$ written as a single log?
A. $\log _5 21$
B. $\log _5 26$
C. $\log _5 30$
D. $\log _5 56$
Answer (1)
Use the logarithm property lo g b x + lo g b y = lo g b ( x y ) to combine the two logarithms.
Apply this property to the given expression: lo g 5 ( 4 × 7 ) + lo g 5 2 = lo g 5 (( 4 × 7 ) × 2 ) .
Simplify the expression inside the logarithm: ( 4 × 7 ) × 2 = 28 × 2 = 56 .
The expression simplifies to lo g 5 56 .
Explanation
Understanding the Problem We are given the expression lo g 5 ( 4 × 7 ) + lo g 5 2 and we want to write it as a single logarithm. To do this, we will use the logarithm property that states lo g b x + lo g b y = lo g b ( x y ) . This property allows us to combine two logarithms with the same base into a single logarithm by multiplying their arguments.
Applying Logarithm Properties Applying the logarithm property, we have: lo g 5 ( 4 × 7 ) + lo g 5 2 = lo g 5 (( 4 × 7 ) × 2 ) Now, we simplify the expression inside the logarithm: ( 4 × 7 ) × 2 = 28 × 2 = 56 So, the expression becomes lo g 5 56 .
Final Answer Therefore, lo g 5 ( 4 ⋅ 7 ) + lo g 5 2 written as a single log is lo g 5 56 .
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 combine and simplify logarithmic expressions is crucial for working with these scales and interpreting the data they provide. For instance, if you have two earthquake readings and want to understand the combined energy released, you would use logarithmic properties to combine the magnitudes.
