Q:

Solve the following logarithmic equations.ln(x^6) = 36

Accepted Solution

A:
Answer:The solution is x = e⁶Step-by-step explanation:Hi there!First, let´s write the equationln(x⁶) = 36Apply logarithm property: ln(xᵃ) = a ln(x)6 ln(x) = 36Divide both sides of the equation by 6ln(x) = 6Apply e to both sidese^(ln(x)) = e⁶x = e⁶The solution is x = e⁶Let´s prove why e^(ln(x)) = xLet´s consider this function:y = e^(ln(x))Apply ln to both sides of the equationln(y) = ln(e^(ln(x)))Apply logarithm property: ln(xᵃ) = a ln(x)ln(y) = ln(x) · ln(e)         (ln(e) = 1)ln(y) = ln(x)Apply logarithm equality rule: if ln(a) = ln(b) then, a = by = xSince y = e^(ln(x)), then x =e^(ln(x))Have a nice day!