Which of the following shows the equation $\log _4(x+6)=2$ rewritten using exponents?
A. $4^{\log _4(x+6)}=2^2$
B. $4^{\log _4(x+6)}=4^4$
C. $\log _4(x+6)=\log _4 16$
D. $\log _4(x+6)=\log _2 16$
Answer (1)
Rewrite the constant 2 as a logarithm with base 4: 2 = lo g 4 4 2 = lo g 4 16 .
Substitute this back into the original equation: lo g 4 ( x + 6 ) = lo g 4 16 .
The equation lo g 4 ( x + 6 ) = 2 rewritten using logarithms is lo g 4 ( x + 6 ) = lo g 4 16 .
Therefore, the answer is lo g 4 ( x + 6 ) = lo g 4 16 .
Explanation
Understanding the Problem We are given the equation lo g 4 ( x + 6 ) = 2 and asked to rewrite it using logarithms.
Expressing 2 as a logarithm with base 4 We need to express the right-hand side of the equation, which is 2, as a logarithm with base 4. We know that 2 = lo g 4 4 2 = lo g 4 16 .
Rewriting the equation Substituting this back into the original equation, we get lo g 4 ( x + 6 ) = lo g 4 16 .
Examples
Logarithmic equations are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, determining the pH levels in chemistry, and modeling population growth in biology. For example, if we want to determine how much the population of a bacteria colony has grown, we can use logarithmic equations to model the growth based on the initial population and the growth rate.
