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Log and Integral Values

by Andrew
(New York)











































a) If (log base 3 of x)*(log base 4 of 2x)*(log base 2x of y) = log base x of x^2 , find the value of y.
b) Find the integral values of x, y, and z that satisfy all of the following equations:
z^x = y^(2x)
2^z = 2*4^x
x+y+z = 16

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Feb 04, 2012
Log and Integral Values
by: Staff


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Part II

b) Find the integral values of x, y, and z that satisfy all of the following equations:

z^x = y^(2x)
2^z = 2*4^x
x+y+z = 16


solve for y in terms of z

z^x = y^(2x)

z^x=(y^2)^x

∴ z=y^2

√z = √(y^2)

√z = y

y = √z



2^z=2^(4x)

∴ z = 4x


since y = √z

y = √(4x)


x + y + z = 16

x + √(4x) + 4x = 16


solve for x

x + 4x + √(4x) = 16

5x + √(4x) = 16

5x - 5x + 2√(x) = -5x + 16

0 + 2√(x) = -5x + 16

2√(x) = -5x + 16

[2√(x)]² = [-5x + 16]²

2² * [√(x)]² = [-5x + 16]²

4x = [-5x + 16]²

4x = 25x² - 160x + 256

4x - 4x = 25x² - 160x - 4x + 256

0 = 25x² - 164x + 256

25x² - 164x + 256 = 0


Applying the quadratic formula:

ax² + bx + c = 0

x = [-b ± √(b² - 4ac)]/(2a)


x₁ = [-(- 164) + √((- 164)² - 4*25*256)]/(2*25)

x₁ = [164 + √((- 164)² - 4*25*256)]/(2*25)

x₁ = [164 + √(26896 - 25600)]/(50)

x₁ = [164 + √(1296)]/(50)

x₁ = [164 + 36]/(50)

x₁ = 200/50

x₁ = 4



x₂ = [164 - 36]/(50)

x₂ = 128/50

x₂ = 2.56


x ∈ {2.56, 4}


Since you now know the values of x, you can use these values to find y and z.


y = √(4x)


for x₁ = 4


y₁ = √(4*4)

y₁ = 4


for x₂ = 2.56


y₂ = √(4*2.56)

y₂ = 2*1.6

y₂ = 3.2



z = 4x


for x₁ = 4

z₁ = 4*4

z₁ = 16



for x₂ = 2.56


z₂ = 4*2.56

z₂ = 10.24




The final answer is:

x₁ = 4
y₁ = 4
z₁ = 16


x₂ = 2.56
y₂ = 3.2
z₂ = 10.24



Thanks for writing.

Staff
www.solving-math-problems.com


Feb 04, 2012
Log and Integral Values
by: Staff


Part I

Question:

by Andrew
(New York)


a) If (log base 3 of x)*(log base 4 of 2x)*(log base 2x of y) = log base x of x^2 , find the value of y.
b) Find the integral values of x, y, and z that satisfy all of the following equations:
z^x = y^(2x)
2^z = 2*4^x
x+y+z = 16



Answer:


a) If (log base 3 of x)*(log base 4 of 2x)*(log base 2x of y) = log base x of x^2 , find the value of y.


Change all logarithms to the same base. For this problem, you can use a base of 10


use this LOGARITHMIC IDENTITY for CHANGING the BASE

log_base a_(b) = [log_base c_(b)] / [log_base c_(a)]


log base 3 of x

log_base 3_(x) = [log_base 10_(x)] / [log_base 10_(3)]


log base 4 of 2x

log_base 4_(2x) = [log_base 10_(2x)] / [log_base 10_(4)]


log base 2x of y

log_base 2x_(y) = [log_base 10_(y)] / [log_base 10_(2x)]


log base x of x^2

log_base x_( x^2) = [log_base 10_( x^2)] / [log_base 10_(x)]





(log base 3 of x)*(log base 4 of 2x)*(log base 2x of y) = log base x of x^2

{[log(x)] / [log(3)]}*{[log(2x)] / [log(4)]}*{ [log(y)] / [log(2x)]} = [log( x^2)] / [log(x)]

[log(2x)] in the numerator cancels with [log(2x)] in the denominator

{[log(x)] / [log(3)]}*{1 / [log(4)]}*{ [log(y)] / 1} = [log( x^2)] / [log(x)]

{[log(x)] / [log(3)]}*{[log(y)] / [log(4)]} = [log( x^2)] / [log(x)]

{[log(x)] / [log(3)]}*{[log(y)] / [log(4)]} = [2log( x)] / [log(x)]

{[log(x)] / [log(3)]}*{[log(y)] / [log(4)]} = 2

{[log(x)] * [log(y)]} / {[log(3)] * [log(4)]} = 2

{[log(x)] * [log(y)]}= 2 * {[log(3)] * [log(4)]}


log(y) * log(x)= 0.5745112369578


Open the following link to see a graphical solution for x and y:


http://www.solving-math-problems.com/images/log-function-graph-A-2012-02-04.png


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