Question

the number of bacteria in an experiment can be represented by $f(x) = 4^x$, where $x$ represents the number of hours. what is $x$ when $f(x) = 64$? a. $x = 60$; in the 60th hour there will be 64 bacteria. b. $x = 3$; in the 3rd hour there will be 64 bacteria. c. $x = 16$; in the 16th hour there will be 64 bacteria. d. $x = 4$; in the 4th hour there will be 64 bacteria.

Explanation:

Step1: Substitute \( f(x) \) with 64

We know that \( f(x) = 4^x \) and \( f(x) = 64 \), so we set up the equation \( 4^x = 64 \).

Step2: Rewrite 64 as a power of 4

We know that \( 4^3 = 64 \) (because \( 4\times4\times4 = 64 \)). So the equation becomes \( 4^x = 4^3 \).

Step3: Solve for \( x \)

Since the bases are the same and the exponential function is one - to - one, if \( a^m=a^n \), then \( m = n \). Here \( a = 4 \), \( m=x \) and \( n = 3 \), so \( x=3 \).

Answer:

B. \( x = 3 \); In the 3rd hour there will be 64 bacteria.