Question

Which of the following inequalities correctly represents the solution to (4ˣ - 2 ⋅ 5²ˣ - 10ˣ > 0)?

a) (x < log(2)/log(2) - log(5))
b) (x > log(2)/log(2) - log(5))
c) (x < log_5/2left(1/2right))
d) (x > log_5/2left(1/2right))

Answer 1

To solve the given inequality (4^x - 2 · 5^(2x) - 10^x > 0), one must manipulate terms to find a common base and potentially use logarithmic properties. The options given in the question do not directly apply to the original inequality without further manipulation. A more precise solution cannot be identified from the options a), b), c), or d) without additional context or steps in algebraic manipulation.

Step-by-step explanation:

The given inequality is (4x - 2 · 52x - 10x > 0). To solve this inequality, we will attempt to manipulate and simplify the terms.

First we notice that the terms 4x and 10x have a common base, which is 2. Recalling that 10 = 2 · 5, we can rewrite 10x as (2 · 5)x or 2x · 5x. This allows us to express the inequality in terms involving common bases:

· (4x - 2(2x+1) - 2x · 5x > 0)

To proceed further, one may attempt to find a common exponential expression or use logarithms to solve the inequality. However, without additional context, information or clarification on the intention behind the given options, it is difficult to identify a precise solution. Therefore, based on the information provided and keeping in mind that the property of logarithms does provide that the logarithm of a division (loga(b/c)) is equal to the difference of logarithms (loga(b) - loga(c)), neither of the options a) nor b) seem to apply directly because they lack the proper expression of the original inequality.

Considering a change of base formula or logarithmic properties might shed some light, but without further steps in the algebraic manipulation of the original inequality, confidently choosing between the given options a), b), c), or d) is not possible based on the data provided.