I tried everthing but,I couldn't solve it. Please help me.the answer at the back is 40960.13
question: solve for x and check your solution.
logx/log2 + logx/log4 + logx/log8 + logx/log16 = 25
THANK YOU FOR YOUR KIND HELP!!!!
Multiply both sides by log16.
log16/log2=4
log16/log4=2
log16/log8=4/3
log16/log16=1
logx(4+2+4/3+1)=25*log16
logx=3*log16
logx=log(16^3)
x=16^3=4096 (check the answer at the back)
To solve the equation logx/log2 + logx/log4 + logx/log8 + logx/log16 = 25, we need to use some logarithmic properties and algebraic manipulation.
Step 1: Rewrite the logarithms in terms of a common base.
- Since the logarithms have different bases (2, 4, 8, and 16), we need to express them in terms of a common base. The most common base used is 10, but we can also choose the base "e", which is the natural logarithm, or we can choose any other base we are comfortable with. For simplicity, let's choose base 2 for all the logarithms in this problem.
- By using the change of base formula, we can rewrite the equation as:
logx/log2 + logx/log2^2 + logx/log2^3 + logx/log2^4 = 25,
Or, equivalently:
logx/1 + logx/2 + logx/3 + logx/4 = 25.
Step 2: Combine the logarithms using the properties of logarithms.
- We can combine the logarithms by using the logarithmic property:
loga + logb = log(ab).
- Applying this property, we can rewrite the equation as:
logx((x/1)(x/2)(x/3)(x/4)) = 25,
Or, equivalently:
logx(x^4/24) = 25.
Step 3: Convert the logarithmic equation to an exponential equation.
- By definition, logarithms and exponentials are inverse operations. Converting the logarithmic equation to an exponential equation helps us solve for x.
- Rewrite the equation in exponential form:
x^4/24 = 2^25,
Or, equivalently:
x^4/24 = 33554432.
Step 4: Solve the exponential equation for x.
- Multiply both sides of the equation by 24 to get rid of the fraction:
x^4 = 24 * 33554432,
Or:
x^4 = 805306368.
Step 5: Take the fourth root of both sides to solve for x.
- Raise both sides of the equation to the power of 1/4 to solve for x:
x = (805306368)^(1/4),
Or:
x ≈ 40960.13.
Step 6: Verify the solution.
- To check if x ≈ 40960.13 is the correct solution, substitute this value back into the original equation and see if it holds true.
- Plug in x = 40960.13 into the original equation and simplify:
log(40960.13)/log2 + log(40960.13)/log4 + log(40960.13)/log8 + log(40960.13)/log16 = 25.
- Calculate the left side of the equation and see if it equals 25.
- If the left side equals 25, then x ≈ 40960.13 is the correct solution to the equation.