Final answer:
To solve the equation 2^2x=5^x-1, we use the natural logarithm to manipulate and solve for x, leading to the exact solution x = ln(5) / (2 * ln(2) - ln(5)).
Step-by-step explanation:
To solve the exponential equation 2^2x=5^x-1, we want to express both sides with the same base if possible. Unfortunately, 2 and 5 are prime numbers and cannot be expressed in terms of each other using integer exponents. Instead, we'll use logarithms to solve for the exponent x. By applying the natural logarithm (ln), we can use its properties to isolate x.
Apply the natural logarithm to both sides: ln(2^2x) = ln(5^x-1).
Use the property of logarithms that allows us to move the exponent to the front: 2x * ln(2) = (x - 1) * ln(5).
Expand the right side: 2x * ln(2) = x * ln(5) - ln(5).
Collect all x terms on one side and constants on the other side.
Factor out x: x(2 * ln(2) - ln(5)) = ln(5).
Divide by the coefficient of x: x = ln(5) / (2 * ln(2) - ln(5)).
Note that we can calculate ln(2) and ln(5) using a calculator, but in our exact solution, we leave the expression in terms of natural logarithms. This is the exact solution for the variable x in the given equation.