Can someone explain how to do this problem?
11^(x+5) = 8e^(8-x)
ln 11^(x+5) = ln 8e^(8-x)
ln 11^(x+5) = ln 8 + ln e^(8-x)
(x+5) ln 11 = ln 8 + (8-x) ln e
(x+5) ln 11 = ln 8 + (8-x)
x ln 11 + x = 8 + ln 8 - 5 ln 11
x(ln11 + 1) = 8 + ln 8 - 5 ln 11
x = (8 + ln 8 - 5 ln 11) / (ln11 + 1)
x ≈ -0.5621
11^(x + 5) = 8e^(8 - x)
11^(x + 5) = 8•e^(8 - x)
=> Apply the natural logarithmic function "In" to both sides, i.e.
ln 11^(x + 5) = ln [8•e^(8 - x)]
=> Consider these rules: In a^b = bIn a & In (a•b) = In a + In b, then by applying them, you'll obtain:
(x + 5) ln 11 = ln 8 + In e^(8 - x)
=> Also, consider this: In e^a = a, then
xIn 11 + 5In 11 = ln 8 + (8 - x)
xIn 11 + 5In 11 = In 8 + 8 - x
=> Pair like terms, i.e.
xln 11 + x = 8 + ln 8 - 5ln 11
=> Factor out x at the left hand side, i.e.
x(ln 11 + 1) = 8 + ln 8 - 5 ln 11
=> Divide both sides by (In 11 + 1), then
x = (8 + ln 8 - 5ln 11)/(ln 11 + 1)
=> Using a scientific calculator, compute the expression at the right hand side on the calculator, then
x â - 0.5621 ...Ans.
Comments
11^(x+5) = 8e^(8-x)
ln 11^(x+5) = ln 8e^(8-x)
ln 11^(x+5) = ln 8 + ln e^(8-x)
(x+5) ln 11 = ln 8 + (8-x) ln e
(x+5) ln 11 = ln 8 + (8-x)
x ln 11 + x = 8 + ln 8 - 5 ln 11
x(ln11 + 1) = 8 + ln 8 - 5 ln 11
x = (8 + ln 8 - 5 ln 11) / (ln11 + 1)
x ≈ -0.5621
11^(x + 5) = 8e^(8 - x)
11^(x + 5) = 8•e^(8 - x)
=> Apply the natural logarithmic function "In" to both sides, i.e.
ln 11^(x + 5) = ln [8•e^(8 - x)]
=> Consider these rules: In a^b = bIn a & In (a•b) = In a + In b, then by applying them, you'll obtain:
(x + 5) ln 11 = ln 8 + In e^(8 - x)
=> Also, consider this: In e^a = a, then
xIn 11 + 5In 11 = ln 8 + (8 - x)
xIn 11 + 5In 11 = In 8 + 8 - x
=> Pair like terms, i.e.
xln 11 + x = 8 + ln 8 - 5ln 11
=> Factor out x at the left hand side, i.e.
x(ln 11 + 1) = 8 + ln 8 - 5 ln 11
=> Divide both sides by (In 11 + 1), then
x = (8 + ln 8 - 5ln 11)/(ln 11 + 1)
=> Using a scientific calculator, compute the expression at the right hand side on the calculator, then
x â - 0.5621 ...Ans.