logx^4+logx^2=6

Question

log₄x+log₂x=6

Answer 1

logx base 4 + logx base 2 = 6

x =16

4^2 = 16, 2^4 = 16

2+4 = 6

Answer 2

Logarithm base b of x is written log_b(x).
By convention, log_10(x) = log(x) and log_e(x) = ln(x).

 

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First problem:

 

log(x^4)+log(x^2) = 6

Use Product Rule of Logs:
log(x^4 * x^2) = 6
log(x^6) = 6

Use Power Rule of Logs:
6 * log(x) = 6
log(x) = 1

Write equivalent exponential equation:
10^1 = x
x = 10

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Second problem:

log_4(x) + log_2(x) = 6

Let log_4(x) = y

Convert to exponential form:
4^y = x
2^(2y) = x

Convert to logarithmic form:
log_2(x) = 2 * log_4(x)

Substitute into original equation:
log_4(x) + 2 * log_4(x) = 6
3 * log_4(x) = 6
log_4(x) = 2

Convert to exponential form:
4^2 = x
x = 16

Answer 3

If it's log(x⁴)+log(x²) = 6, it's easy.

 

Since log(a·b) = log(a) + log(b),

 

log(xⁿ) = n·log(x)

 

So:

 

log(x⁴)+log(x²) = 6

4·log(x)+2·log(x) = 6

6·log(x) = 6

log(x) = 1

10^log(x) = x = 10¹ 

=> x = 10

Answer 4

log X^4 + log X^2 = 6

 

   Log ( X^4 . log X^2 ) = 6

 

    Log X^6 = 6

 

      X^ 6 = 10 ^6

 

     X = 10

 

 Using Properties of Log function.

 

   Log ( A . B ) = Log A + Log B ,      and basic definition of log function:    A = a ^ b

 

                                                                                                        b = Log A _b

 

2) 

 log 4X +log 2X = 6

 

  Log ( 4X . 2X) = 6

 

      Log 8X^2 = 6

 

        8X^2 = 10^6

      

        X^2 = 10^6/8

 

         X = 10^3/ (2√2) = 1000( √2) /2 = 500 √2   

 

 

Answer 5

X=16. Look for perfect squares with factors 2 and 4. 4 is the smallest but doesn't work here.