Respuesta :
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) - 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]
