To solve the equation 5^x = 4^(x+1), you are correct that finding a common base can be helpful. However, when you attempted to use 20 as the common base, you made an error in the exponent.
The correct approach is to use logarithms to convert the exponential equation into a form that is easier to solve. In this case, we can use the natural logarithm (log base e), denoted as ln.
Step 1: Take the natural logarithm of both sides of the equation:
ln(5^x) = ln(4^(x+1))
Step 2: Use the logarithmic property that allows you to bring the exponent down:
x * ln(5) = (x+1) * ln(4)
Step 3: Expand the equation using the distributive property:
x * ln(5) = x * ln(4) + ln(4)
Step 4: Move all the terms with x to one side of the equation and move the constant term to the other side:
x * ln(5) - x * ln(4) = ln(4)
Step 5: Factor out the x:
x * (ln(5) - ln(4)) = ln(4)
Step 6: Divide both sides of the equation by (ln(5) - ln(4)):
x = ln(4) / (ln(5) - ln(4))
Using a calculator, calculate ln(4), ln(5), and perform the division to find the numerical value of x.
Note: The exact value of x may be difficult to find without a calculator, so it is recommended to use a calculator to approximate the solution.