Number of real solutions of the equation #log_10(-x) = sqrt(log_10sqrt(x^2))#?

Question

HIX Tutor · Accepted Answer

undefined To determine the number of real solutions of the equation log_10(-x) = sqrt(log_10sqrt(x^2)), we can follow these steps:

Step 1: Start by analyzing the given equation:
   - log_10(-x) = sqrt(log_10sqrt(x^2))

Step 2: Notice that the equation involves the logarithm of a negative number (-x).
   - Logarithms are only defined for positive numbers, so the equation is not defined for all values of x.

Step 3: Let's examine the inside of the logarithm log_10sqrt(x^2):
   - The square root of a number is always non-negative, including zero.
   - The square root of x^2 will only be zero if x is zero, and in that case, the given equation is also not defined since we can't take the logarithm of zero.

Step 4: Based on the analysis from steps 2 and 3, we can conclude that there are no real solutions for the equation log_10(-x) = sqrt(log_10sqrt(x^2)).

In summary, this equation does not have any real solutions.

Ariana Alvarez · Answer

See below.

#sqrt(x^2) = abs x# then #log_10 (-x) = sqrt(log_10 absx)# now calling #y = -x# #log_10 y = sqrt(log_10 abs y)# has sense only for #y= 1# or #y = 10# then the solutions are for #x = {-1, -10}#

Greyson Brown · Answer

undefined The equation log₁₀(-x) = √(log₁₀(√(x²))) has no real solutions.