College Algebra Problems With Answers
sample 3 : Exponential and Logarithmic Functions
College algebra problems on logarithmic and exponential function with answers, are presented along with solutions are at the bottom of the page.
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Let the logarithmic
function f be defined by f(x) = 2ln(2x - 1).
a) Find the domain of f.
b) Find vertical asymptote of the graph of f.
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Let the exponential function h be defined by h(x) = 2 + ex
a) Find the range of h.
b) Find the horizontal asymptote of the graph of h.
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The population of city A changes according to the exponential function
A(t) = 2.9 (2) 0.11 t (millions)
and the population of city B changes according to the exponential function
B(t) = 1.7 (2) 0.17 t (millions)
where t = 0 correspond to 2009.
a) Which city had larger population in 2009?
b) When will the sizes of the populations of the two cities be equal?
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Find the inverse of the logarithmic function f defined by f(x) = 2 Log5 (2x - 8) + 3
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Find the inverse of the exponential function h defined by h(x) = - 2*3-3x + 9 - 4
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Solve the logarithmic equation defined by
ln(2x - 2) + ln(4x - 3) = 2 ln(2x)
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A, B and k in the exponential function f given by
f(x) = A ek x + B
are constants. Find A, B and k if f(0) = 1 and f(1) = 2 and the graph of f has a horizontal asymptote y = -4.
Answers to the Above Questions
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- solve 2x - 1 > 0 to find domain: x > 1/2
- solve 2x - 1 v = 0 to find vertical asymptote: x = 1/2
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- range of h: (2, +infinity)
- horizontal asymptote: y = 2
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- A(0) = 2.9 millions , B(0) = 1.7 millions, city A had larger population.
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solve 2.9 (2) 0.11 t = 1.7 (2) 0.17 t, to find t.
take ln of both sides of the equation
ln[ 2.9 (2) 0.11 t ] = ln[ 1.7 (2) 0.17 t ]
ln(2.9) + 0.11t ln(2) = ln(1.7) + 0.17t ln(2)
solve for t:
t = (ln1.7 - ln2.9) / (0.11ln2 - 0.17ln2) = 13 (approximated to the nearest unit)
The size of the two populations will be the same in 2009 + 13 = 2022.
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solve the equation: x = 2 Log5 (2y - 8) + 3 for y to obtain the inverse of the function.
f -1(x) = (1/2) 5 (x-3)/2 + 4
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solve the equation: x = - 2*3-3y + 9 - 4 for y to obtain the inverse of the function.
h -1(x) = (-1/3)Log3 [(x+4)/-2] + 3
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Rewrite the given equation as follows
ln(2x - 2)(4x - 3) = ln(2x)2
The above gives the algebraic equation
(2x - 2)(4x - 3) = (2x)2
Solve the above quadratic equation for x
x = 3 and x = 1/2
Check the two values of x and only x = 3 is a solution to the given equation.
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Horizontal asymptote y = - 4 gives B = - 4.
f(0) = A + B = 1
which gives A = 5 since B = - 4
f(1) = 5ek - 4 = 2
solve for k to obtain: k = ln(6 / 5) More References and links
logarithmic functions
exponential functions
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