Question

1 mark for review the functions f and g are given by f(x) = 4^(3x - 1) and g(x) = 8^(x + 4). when solving the equation f(x) = g(x), the functions can be rewritten in equivalent forms so that the equation can be solved without the use of technology. which of the following are equivalent definitions of f and g that aid in solving f(x) = g(x) without the use of technology? a f(x) = 2^(log₂4(3x - 1)) and g(x) = 2^(log₂8(x + 4)) b f(x) = 2^(log₂8(3x - 1)) and g(x) = 2^(log₂4(x + 4)) c f(x) = 4^(log₂4(3x - 1)) and g(x) = 8^(log₂8(x + 4)) d f(x) = 2·4^(log₂4(3x - 1)) and g(x) = 8^(log₂8(x + 4))

Explanation:

To solve \( f(x) = g(x) \) where \( f(x) = 4^{(3x - 1)} \) and \( g(x) = 8^{(x + 4)} \), we can use the property of exponents and logarithms: \( a^b = 2^{\log_2 a^b} = 2^{b\log_2 a} \) (since \( \log_a a^b = b \) and \( a^b = 2^{\log_2 a^b} \) by the change - of - base formula for logarithms).

Step 1: Rewrite \( f(x) \)

For \( f(x)=4^{(3x - 1)} \), we know that \( 4 = 2^2 \), and we can also use the formula \( a^b=2^{\log_2 a^b}=2^{b\log_2 a} \).
We know that \( \log_2 4=\log_2 2^2 = 2 \), and \( 4^{(3x - 1)}=2^{\log_2 4^{(3x - 1)}} \) (because for any positive number \( a
eq1 \) and \( N>0 \), \( a = 2^{\log_2 a} \), so \( a^b=(2^{\log_2 a})^b = 2^{b\log_2 a} \), here \( a = 4 \), \( b=(3x - 1) \)). By the power rule of logarithms \( \log_a M^n=n\log_a M \), \( \log_2 4^{(3x - 1)}=(3x - 1)\log_2 4 \). So \( f(x)=2^{(\log_2 4(3x - 1))} \)

Step 2: Rewrite \( g(x) \)

For \( g(x)=8^{(x + 4)} \), since \( 8=2^3 \), using the same property \( a^b = 2^{\log_2 a^b} \) with \( a = 8 \) and \( b=(x + 4) \).
We know that \( \log_2 8=\log_2 2^3=3 \), and \( 8^{(x + 4)}=2^{\log_2 8^{(x + 4)}} \). By the power rule of logarithms, \( \log_2 8^{(x + 4)}=(x + 4)\log_2 8 \). So \( g(x)=2^{(\log_2 8(x + 4))} \)

After rewriting \( f(x) \) and \( g(x) \) in terms of base 2, we can set the exponents equal to each other (since if \( 2^m=2^n \), then \( m = n \)) and solve for \( x \) without using technology. The equivalent forms are given in option A.

Answer:

A. \( f(x)=2^{(\log_{2}4(3x - 1))} \) and \( g(x)=2^{(\log_{2}8(x + 4))} \)