To the nearest hundredth, what is the value of x in the equation 5^(3x) = 37?

To find the value of ( x ) in the equation ( 5^{3x} = 37 ), we need to isolate ( x ). This involves using logarithms, as they allow us to bring down the exponent in an exponential equation.

First, we can take the logarithm of both sides of the equation. Using natural logarithms (though common logarithms would also work), we have:

[

\ln(5^{3x}) = \ln(37)

]

Applying the power rule of logarithms, which states that ( \ln(a^b) = b \cdot \ln(a) ), we can rewrite the left side:

[

3x \cdot \ln(5) = \ln(37)

]

To isolate ( x ), divide both sides by ( 3\ln(5) ):

[

x = \frac{\ln(37)}{3\ln(5)}

]

Next, we can evaluate this expression using a calculator to get numeric values for ( \ln(37) ) and ( \ln(5) ):

1. Calculate ( \ln(37) ) which is approximately ( 3.6109 \