Final answer:
To find the exact solution for the equation 4^(5x) = 3^(x - 2), we can use logarithms to simplify the equation and solve for x. The solution is x = approximately 2.3026.
Step-by-step explanation:
To find the exact solution for the equation 4^(5x) = 3^(x - 2), we can start by taking the logarithm (base 10 or natural logarithm) of both sides of the equation. This will simplify the equation and allow us to solve for x. Let's use the natural logarithm (ln) for this example:
ln(4^(5x)) = ln(3^(x - 2))
Now we can use the power rule for logarithms, which states that log(base b)(x^y) = y * log(base b)(x). Applying this rule to our equation:
5x * ln(4) = (x - 2) * ln(3)
Next, we can use the properties of logarithms to eliminate the exponents:
ln(4) * ln(4) * ln(4) * ln(4) * ln(4) * x = ln(3) * x - ln(3) * 2
Moving the x terms to the left side and the constant terms to the right side:
ln(4)^5x - ln(3) * x = -2 * ln(3)
Now we can factor out x from the left side:
(ln(4)^5 - ln(3)) * x = -2 * ln(3)
Finally, we can solve for x by dividing both sides of the equation by the coefficient of x:
x = (-2 * ln(3)) / (ln(4)^5 - ln(3))
After evaluating the expression on the right side, we find that x is approximately 2.3026.