Could someone help me with this question?
5^(2x) - 4^(5^x) = 12
I got up to log (5^x)= log(-4)
but it isnt correct since it can't be negative.
I don't know if this will show up properly, but do you mean
52x - 45x = 12 ?
How did you get log(5^x) = log(-4) ??
Yes
to get that i did:
(5^x)^2 - 4(5^x) =12
5^x(1-4) = 12
-3(5^x)= 12
log(5^x) = log(-4)
If you could show me the correct way to do it, that would be great! thanks!
ah, but now you changed the question.
In the original you had 4 raised to the (5 raised to the x) in the middle term
Now you have 4(5^x)
This last version makes it easy, the first version would be a nightmare.
If it is
(5^x)^2 - 4(5^x) =12
then let's let y = 5^x
our equation now becomes
y^2 - 4y = 12 which is a nice quadratic
y^2 - 4y - 12 = 0
(y-6)(y+2) = 0
so y = 6 or y = -2
then 5^x = 6
log (5^x) = log6
xlog5 = log6
x = log6/log5 = 1.11328
or 5^x = -2
xlog5 = log(-2) which is undefined, thus no solution for that part
so x = appr. 1.113
The original question was what i needed to solve, but with the way i changed it, is it correct?
My answer, if substituted into the second version of the equation, satisfies it.
To solve the equation 5^(2x) - 4^(5^x) = 12, we need to use some algebraic manipulation and properties of logarithms. The approach you mentioned using logarithms is a good start, but it seems like there might have been a mistake in your calculation.
Let's start again and go step by step:
1. Start with the original equation: 5^(2x) - 4^(5^x) = 12.
2. The base of the first term is 5, so let's isolate it by subtracting 12 from both sides: 5^(2x) - 4^(5^x) - 12 = 0.
3. Notice that the base of the second term is 4, which is 2 raised to the power of 2. We can rewrite the second term using this information: 5^(2x) - (2^2)^(5^x) - 12 = 0.
4. Apply the exponent rule (a^b)^c = a^(b*c) to simplify the second term: 5^(2x) - 2^(2*(5^x)) - 12 = 0.
5. Now, let's apply a substitution to make it easier to work with: let y = 5^x. Rewriting the equation using this substitution, we have: 5^(2x) - 2^(2y) - 12 = 0.
6. Now, the equation becomes: 5^(2x) - 2^(2y) = 12.
At this point, we have a system of equations:
- First equation: 5^(2x) - 2^(2y) = 12.
- Second equation: y = 5^x.
To solve this system, we have a few options. One approach is to use a numerical method, such as graphing both equations and finding their intersection point, or using an iterative method like Newton's method. Another approach would be to transform the equations so that they are in terms of a single variable, either x or y, and then solve.
Unfortunately, in this case, the equation involves both exponential and logarithmic functions, so it does not have a simple algebraic solution. It requires the use of numerical methods or approximation techniques to solve for the values of x and y that satisfy the equation.