Question

Log16(31x−1)−log16x=0.

Answer 1

Ответ:

Пошаговое объяснение:

Answer 2

Ответ:

Пошаговое объяснение:

Answer 3

To solve the logarithmic equation log₁₆(31x⁻¹) - log₁₆(x) = 0, we can use logarithmic properties to simplify and solve for x. Here's the step-by-step solution:

Step 1: Apply the quotient rule of logarithms, which states that logₐ(b) - logₐ(c) = logₐ(b/c):

log₁₆(31x⁻¹) - log₁₆(x) = log₁₆(31x⁻¹ / x)

Step 2: Simplify the expression within the logarithm:

log₁₆(31/x)

Step 3: Apply the change of base formula to rewrite the logarithm in a common base, such as base 10 or base e (natural logarithm):

log₁₆(31/x) = logₓ(31/x) / logₓ(16)

Step 4: Since the equation is given in terms of x, you can now solve for x by setting the expression equal to 0:

logₓ(31/x) / logₓ(16) = 0

Step 5: The only way the logarithmic expression can be equal to 0 is if the numerator is 0:

logₓ(31/x) = 0

Step 6: Convert the logarithmic equation into an exponential equation:

x⁰ = 31/x

Step 7: Any non-zero number raised to the power of 0 is 1, so:

1 = 31/x

Step 8: Solve for x:

x = 31

So, the solution to the equation log₁₆(31x⁻¹) - log₁₆(x) = 0 is x = 31.