solve the ff indical equation: 2^(2x+8)-32^(2x+1)=0
2.4^x=0.59
are you saying that
2^(2x+8) = - [ 2^5]^(2x+1)
then
2x+8 = -5 (2x+1)
2 x + 8 = -10 x - 5
12 x = - 13
x = -13/12
2^(2x+8)-32^(2x+1)=0
2^(2x+8) = 32^(2x+1)
2^(2x+8) = 2^(10x+5)
2x+8 = 10x+5
8x = 3
x = 3/8
2.4^x=0.59
x log2.4 = log0.59
x = log0.59/log2.4 = -0.6
To solve the first equation, let's break it down step by step:
Step 1: Simplify the equation.
We know that 32 can be written as 2^5. So, rewrite the equation as:
2^(2x+8) - (2^5)^(2x+1) = 0.
Step 2: Simplify the exponents.
We can simplify (2^5)^(2x+1) by applying the exponent rule:
(2^5)^(2x+1) = 2^(5*(2x+1)) = 2^(10x + 5).
Step 3: Use the properties of exponents.
Apply the property of exponents that states a^(b+c) = a^b * a^c:
2^(2x+8) - 2^(10x + 5) = 0.
Step 4: Set the bases equal to each other.
Since both terms have the same base (2), set the exponents equal to each other:
2x + 8 = 10x + 5.
Step 5: Solve for x.
Move the variables to one side and the constants to the other side of the equation:
2x - 10x = 5 - 8,
-8x = -3.
Divide both sides of the equation by -8:
x = (-3) / (-8).
Simplifying the fraction gives:
x = 3/8.
Therefore, the solution to the first equation is x = 3/8.
Moving on to the second equation:
Step 1: Take the logarithm of both sides.
We can use the logarithm function to solve this equation. Taking the logarithm (base 2.4) of both sides, we get:
log base 2.4(2.4^x) = log base 2.4(0.59).
Step 2: Apply the logarithm property.
The logarithm of a number raised to a power can be rewritten as the product of that power times the logarithm of the number. Applying this property, the equation becomes:
x * log base 2.4(2.4) = log base 2.4(0.59).
Step 3: Evaluate the logarithms.
The logarithm of the base (2.4) to itself is 1, so the equation simplifies to:
x = log base 2.4(0.59).
Step 4: Use a calculator to find the approximate value.
Using a calculator, you can evaluate the logarithm of 0.59 with a base of 2.4, giving you the approximate value of x.
Please note that the approximate value will depend on the accuracy of your calculator and the number of decimal places you choose to round to.
I hope this helps! Let me know if you have any further questions.