solving exponential equations with exponents on each side
5^x =4^(x+1)
They don't have common bases and now I am lost I thought I could just come up with a common base (in this case 20) and change the equation to
20^4x =20^(5x+5)
However I am not getting the correct answer!
Major Error !
How did you mathemagically change 5^x to 20^4x ?
Did you do something like (5^x)(4^4) = 20^4x ??
If so then something like this should work :
(3^5)(4^2) = 12^10 which is certainly false
You can only do this type of question if you know logs.
take the log of both sides
log(5^x) = log(4^(x+1))
using the rules of logs,
x(log5) = (x+1)(log4)
x(log5) = xlog4 + log4
xlog5 - xlog4 = log4
x(log5 - log4) = log4
x = log4/(log5 - log4)
= 6.21257
Thanks! I'm sorry for sooo many questions but I took this as a TERM class and so I've had 4 weeks to learn basically 1/2 the book with the final on Monday and he never really got time to lecture on Chapter 4 so I am basically trying to learn it on my own with your help and the help of videos on line!
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.