Final answer:
To solve the equation 7⁴x = 3.9, take the logarithm of both sides, bring down the exponent, and divide by ln(7) to isolate x, giving the exact solution x = ln(3.9) / (4 * ln(7)).
Step-by-step explanation:
To solve the equation 7⁴x = 3.9, first, take the logarithm of both sides of the equation to extract the exponent. You can use either the natural logarithm (ln) or the common logarithm (log), but for consistency, let's use the natural logarithm:
ln(7⁴x) = ln(3.9)
Now, use the property of logarithms that allows us to bring down the exponent:
4x * ln(7) = ln(3.9)
Divide both sides by ln(7) to solve for x:
x = ln(3.9) / (4 * ln(7))
This gives the exact solution for x. You can use a calculator to find a decimal approximation, but the question asks for an exact answer, which is the expression above.