If 4^x-4^x-1=24 find the value of x
If you mean 4^x - 4^(x-1) = 24
then If you let
u = 4^(x-1)
then you have
4u-u = 24
u = 8
4^(x-1) = 8
x-1 = 3/2
x = 5/2
Thanks
To solve the equation 4^x - 4^(x-1) = 24 and find the value of x, let's break it down step by step:
Step 1: Simplify the equation on the left side.
4^x - 4^(x-1) can be rewritten as (4^x) - (4^x / 4^1) = 24.
Step 2: Simplify the exponent on the right side.
4^1 can be further simplified as 4.
Now the equation becomes (4^x) - (4^x / 4) = 24.
Step 3: Combine the terms with a common base.
Apply the exponent rule: a^m / a^n = a^(m-n).
(4^x) - (1/4)(4^x) = 24.
Step 4: Find a common denominator and combine the fractions.
(4^x) - (1/4)(4^x) can be written as (4^x) - (4^x / 4) = 24.
Multiplying the second term by 1 will help us find the common denominator.
(4^x) - (4^x / 4) = 24 * 1.
Step 5: Simplify the equation further.
Multiply 24 by 4, which gives us 96.
(4^x) - (4^x / 4) = 96.
Step 6: Combine the terms with a common base.
The term (4^x / 4) can be further simplified as (4^(x-1)).
(4^x) - (4^(x-1)) = 96.
Step 7: Set the equation equal to zero.
Now, we rearrange the equation by moving 96 to the other side:
(4^x) - (4^(x-1)) - 96 = 0.
Step 8: Factor out the common term.
Common term here is 4^(x-1):
(4^(x-1))(4 - 1) - 96 = 0.
Simplifying further:
(4^(x-1))(3) - 96 = 0.
Step 9: Divide both sides by 3.
(4^(x-1))(3) - 96 = 0 / 3.
(4^(x-1))(3) = 32.
Step 10: Divide both sides by 3 to isolate the exponential term.
(4^(x-1))(3/3) = 32/3.
(4^(x-1)) = 32/3.
Step 11: Solve for (x-1) using logarithms.
Take the logarithm (base 4) of both sides:
log4(4^(x-1)) = log4(32/3).
Applying the logarithm property: logb(b^x) = x.
(x-1) = log4(32/3).
Step 12: Evaluate the logarithm using a calculator.
Using a calculator, we find that log4(32/3) ≈ 2.833.
Step 13: Solve for x.
x - 1 = 2.833.
Add 1 to both sides:
x = 2.833 + 1.
Final Result:
x = 3.833 (rounded to three decimal places)
Therefore, the value of x is approximately 3.833.