Question

question 1: solve the exponential equation.
$81^{2x} = 3$
$x = 8$ option 1
$x = \frac{1}{8}$ option 2
$x = \frac{3}{16}$
$x = \frac{1}{16}$

Explanation:

Step1: Express 81 as a power of 3

Since \( 81 = 3^4 \), we can rewrite the left - hand side of the equation \( 81^{2x}=3 \) as \( (3^4)^{2x} \).
According to the power - of - a - power rule \((a^m)^n=a^{m\times n}\), we have \((3^4)^{2x}=3^{4\times2x}=3^{8x}\). So the equation becomes \( 3^{8x}=3^1 \).

Step2: Solve for x

If two exponential expressions with the same base are equal, then their exponents must be equal. That is, if \( a^m=a^n \) (where \( a>0,a
eq1 \)), then \( m = n \).
In our equation \( 3^{8x}=3^1 \), since the base \( a = 3\) (which satisfies \( 3>0 \) and \( 3
eq1 \)), we can set the exponents equal to each other: \( 8x=1 \).
To solve for \( x \), we divide both sides of the equation \( 8x = 1\) by 8. So \( x=\frac{1}{8} \).

Answer:

Option 2. \( x=\frac{1}{8} \)