Question

If log 4 16 = log x 36, find x.

Answer 1

Answer:

x = 6

Step-by-step explanation:

Given the logarithmic equation:

$\displaystyle{\log_4 16 = \log_x 36}$

The left side can be simplified to 2 because:

$\displaystyle{\log_4 16 = x}\\\\\displaystyle{4^x = 16}$

And we know that if x = 2 then it'd be true for the equation: So the left side is simplified to 2:

$\displaystyle{2 = \log_x 36}$

Convert logarithm to exponential where:

$\displaystyle{\log_a b = n \to a^n = b}$

Apply the property:

$\displaystyle{\log_x 36 = 2 \to x^2 = 36}\\\\\displaystyle{x^2=36}$

Solve the quadratic equation for x, square root both sides:

$\displaystyle{\sqrt{x^2}=\sqrt{36}$

Cancel square and add plus-minus to right side:

$\displaystyle{x=\pm \sqrt{36}}\\\\\displaystyle{x=\pm 6}$

However, for logarithm function, it's defined that the base cannot be negative. Therefore, -6 is an extraneous solution while x = 6 is the only valid solution.

Therefore, x = 6.