Question

Log 4 x - log 16 x=1\4

Answer 1

Log4 x - 1/2log4 x = 1/4
log4 x - log4 x^1/2 = 1/4
log4 (x/x^1/2)=1/4
log4 x^1/2=1/4
4^1/4=x^1/2
2^(2*1/4)=x^1/2
2^1/2=x^1/2
x=2

Answer 2

To solve the logarithmic equation log₄x - log₁₆x = 1/4, we can use the logarithmic properties to simplify and solve for x.

First, let's apply the quotient rule of logarithms, which states that logₐb - logₐc = logₐ(b/c):

log₄x - log₁₆x = log₄x - (log₄x / log₄16) = log₄x - (log₄x / 2)

Now we have:

log₄x - (log₄x / 2) = 1/4

To simplify the equation further, let's find a common denominator:

(2 * log₄x - log₄x) / 2 = 1/4

(log₄x) / 2 = 1/4

Multiplying both sides by 2:

log₄x = 1/2

Now, we can rewrite the equation in exponential form:

x = 4^(1/2)

Simplifying the right side:

x = 2

Therefore, the solution to the equation log₄x - log₁₆x = 1/4 is x = 2.