Solve for x (log)

Question

log x^log x + log x^2 =  8

Answer 1

logx^logx+ logx² = 8

• logxⁿ = n(logx)
• So bring the exponents down in front of the base

(logx)(logx) + 2logx = 8

• To keep this from getting confusing, substitute y for logx

(y)(y) + 2y = 8

y² + 2y = 8

• Subtract 8 from each side and then factor

y² + 2y - 8 = 0

• Find the common factors

Prod      Sum

-8•1      -8 + 1 = -7

-4•2      -4 + 2 = -2

-2•4      -2 + 4 = 2

• 2y = -2y + 4y so we can substitute this in the equation

y² - 2y + 4y - 8 = 0

• Now group and factor

y² - 2y   +   4y - 8    = 0

y(y-2)    +   4(y-2)    = 0

(y+4)(y-2) = 0

• Then solve for y

(y+4)(y-2)/(y-2) = 0/(y-2)      (y+4)(y-2)/(y+4) = 0/(y+4)

y + 4 = 0                                y - 2 = 0

y + 4 - 4 = 0 - 4                      y - 2 + 2 = 0 + 2

y = -4                                     y = 2

• Now we have two possibilities
• We have to substitute logs back into y

logx = -4          logx = 2

• logx = n is actually log₁₀x = n where 10 is the base and n is the exponent, so x = 10ⁿ

x = 10^-4           x = 10²

 

x = 10^-4 and 10²

• Because this one was complicated, we really need to check our answers to see if they're both correct.
• We'll start with 0.0001

logx^logx+ logx² = 8

log(10^-4)^log(0.0001) + log(10^-4)² = 8

• In your calculator log(0.0001) = -4
• (x^A)^B = x^AB so we multiply the exponents
• (10^-4)² = 10^(-4)(2) = 10^-8

log(10^-4)^-4 + log(10^-8) = 8

• (10^-4)^-4 = 10^(-4)(-4) = 10¹⁶
• In you calculator log(10^-8) = -8

log10¹⁶ + (-8) = 8

• In your calculator log10¹⁶ = 16

16 - 8 = 8

8 = 8 √

 

Our answer is x = 10^-4 and 10² or x = 0.0001 and 100

Answer 2

Hi Lawson;

log x^log x + log x^2 = 8

Because log x is the exponential of x, the first element of the equation can be rewritten as...

(log x)(log x)

Because the 2 is the exponential of x, the second element of the equation can be rewritten as...

2log x

[(log x)(log x)]+(2log x)=8

Let's subtract 8 from both sides...

[(log x)(log x)]+(2log x)-(8)=0

For the FOIL...

FIRST must be (log x)(log x).

OUTER and INNER must add-up to 2log x.

LAST must be (8)(1) or (1)(8) or (2)(4) or (4)(2), and one number must be negative to render the result of -8...

[(log x)+4][(log x)-2]=0

Either or both parenthetical equation(s) must equal zero...

(log x)+4=0

log x=-4

The logarithm is to the base 10.  This can also be written as...

log₁₀x=-4

Let's move the 10 over...

x=10^-4

(log x)-2=0

log₁₀x=2

x=10²=100

Answer 3

 logx ^ (logx ) + log X^2 = 8

 

   X = 10 ^ m

 

   Log X = m     log X^2 = 2 Log X

 

      m^m + 2 m = 8

 

        Try m=2

          2 ^2 + 2( 2) =8

 

     X = 10^2 = 100