$\log \left(10^{6 x}\right)=?$
A. $x$
B. $10^6$
C. 60
D. $6 x$
Answer (2)
Recognize that lo g implies a base of 10.
Apply the logarithmic property lo g b ( b y ) = y .
Simplify lo g ( 1 0 6 x ) to 6 x .
The final answer is 6 x .
Explanation
Understanding the Problem We are asked to simplify the expression lo g ( 1 0 6 x ) . This involves understanding the properties of logarithms, particularly how logarithms and exponentiation with the same base interact.
Identifying the Base The logarithm, written as lo g , without an explicitly specified base, is assumed to be the common logarithm, which has a base of 10. Thus, lo g ( 1 0 6 x ) is equivalent to lo g 10 ( 1 0 6 x ) .
Applying the Logarithmic Property A key property of logarithms states that lo g b ( b y ) = y . In simpler terms, the logarithm base b of b raised to the power of y is just y . Applying this property to our expression, where b = 10 and y = 6 x , we get lo g 10 ( 1 0 6 x ) = 6 x .
Final Answer Therefore, the simplified expression is 6 x .
Examples
Logarithms are incredibly useful in many real-world scenarios. For example, they are used to measure the magnitude of earthquakes on the Richter scale. The formula is M = lo g 10 ( A ) , where M is the magnitude and A is the amplitude of the seismic waves. Also, logarithms are used in chemistry to measure pH levels, where p H = − lo g 10 [ H + ] , with [ H + ] being the concentration of hydrogen ions. Understanding logarithms helps us quantify and understand phenomena that vary over a wide range of values.
The logarithmic expression lo g ( 1 0 6 x ) simplifies to 6 x using the property of logarithms that states lo g b ( b y ) = y . Therefore, the correct answer is 6 x .
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