Question

d.) $4^{5x - 3}=3^{2(log_{3}4)x}$

Explanation:

Step1: Use logarithm properties

First, recall that \( a^{\log_a b}=b \) and \( \log_a b=\frac{\ln b}{\ln a} \). Also, note that \( 3^{2(\log_3 4)x}=(3^{\log_3 4})^{2x} \). Since \( 3^{\log_3 4} = 4 \) (by the property \( a^{\log_a c}=c \)), the right - hand side becomes \( 4^{2x} \). So the equation \( 4^{5x - 3}=3^{2(\log_3 4)x} \) is equivalent to \( 4^{5x-3}=4^{2x} \).

Step2: Solve for x

If \( a^m=a^n \) (where \( a>0,a
eq1 \)), then \( m = n \). For \( a = 4\), we have the equation \( 5x-3=2x \).
Subtract \( 2x \) from both sides: \( 5x-2x-3=2x - 2x \), which simplifies to \( 3x-3 = 0 \).
Add 3 to both sides: \( 3x-3 + 3=0 + 3 \), so \( 3x=3 \).
Divide both sides by 3: \( x = 1 \).

Answer:

\( x = 1 \)