What is the solution for x to three decimal places in the equation 9^(3x) = 45?

To solve the equation ( 9^{3x} = 45 ), it is helpful to first express the left-hand side in terms of base 3, since ( 9 ) can be rewritten as ( 3^2 ). This gives us:

[

(3^2)^{3x} = 3^{6x}

]

Now, we can rewrite the equation as:

[

3^{6x} = 45

]

Next, we take the logarithm of both sides. Using the natural logarithm (though any logarithmic base works), we can write:

[

\ln(3^{6x}) = \ln(45)

]

By using the power property of logarithms, this simplifies to:

[

6x \cdot \ln(3) = \ln(45)

]

Now, we can isolate ( x ) by dividing both sides by ( 6 \ln(3) ):

[

x = \frac{\ln(45)}{6 \ln(3)}

]

At this point, it’s necessary to calculate the values of ( \ln(45) ) and ( \ln(3) ). Using approximate