What is the value of x, to three decimal places, in the equation 4^(2x) = 60?

To find the value of ( x ) in the equation ( 4^{2x} = 60 ), we start by rewriting the left side of the equation with a base of 2, since ( 4 ) can be expressed as ( 2^2 ). Thus, the equation becomes:

[

(2^2)^{2x} = 60

]

This simplifies to:

[

2^{4x} = 60

]

Next, to solve for ( x ), we take the logarithm of both sides. Using the natural logarithm (though common logarithm would yield the same result), we get:

[

\ln(2^{4x}) = \ln(60)

]

Applying the power rule of logarithms, we can pull down the exponent:

[

4x \cdot \ln(2) = \ln(60)

]

Now, to isolate ( x ), we divide both sides by ( 4 \cdot \ln(2) ):

[

x = \frac{\ln(60)}{4 \cdot \ln(2)}

]

Using a calculator to find the values of the logarithms, we