# 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

### Comments for Log and Integral Values

 Feb 04, 2012 Log and Integral Values by: Staff ------------------------------------------------ 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 ------------------------------------------------