MATH SOLVE

3 months ago

Q:
# Solve the following logarithmic equation: In(x +31)-In(4-3x)-5In2 0 x = 2 1 points x= 0 x-0.5 β x=0.25 None of the above to save all

Accepted Solution

A:

Answer:The solution is [tex]x = 1[/tex]Step-by-step explanation:We have the following logarithmic properties:[tex]ln a + ln b = ln ab[/tex][tex]ln a - ln b = ln \frac{a}{b}[/tex][tex]n ln a = ln a^{n}[/tex]We have the following logarithmic equation:[tex]ln(x + 31) - ln (4-3x) - 5 ln 2 = 0[/tex]Lets simplify, and try to find properties.[tex]ln(x + 31) - (ln (4-3x) + 5 ln 2) = 0[/tex][tex]ln(x + 31) - (ln (4-3x) + ln 2^{5}) = 0[/tex][tex]ln(x + 31) - (ln (4-3x) + ln 32) = 0[/tex][tex]ln(x + 31) - Β ln 32*(4-3x) = 0[/tex][tex]ln(x+31) - ln (128 - 96x) = 0[/tex][tex]ln \frac{x + 31}{128 - 96x} = 0[/tex]To eliminate the ln, we apply the exponential to both sides, since e and ln are inverse operations.[tex]e^{ln \frac{x + 31}{128 - 96x}} = e^{0}[/tex][tex]\frac{x + 31}{128 - 96x} = 1[/tex][tex]x + 31 = 128 - 96x[/tex][tex]97x = 97[/tex][tex]x = \frac{97}{97}[/tex][tex]x = 1[/tex]The solution is [tex]x = 1[/tex]