2.13 Exponential and logarithmic equations and inequalities
State the properties of exponents.
State the properties of logarithms.
Explain how to solve an exponential equation.
What is the general form of an exponential function and its transformations?
How do you find the inverse of an exponential function?
Solve the following exponential equations:
\left(\sqrt{6}\right)^{y} = 36
\left(\sqrt{2}\right)^{k} = 0.5
9^{y} = 27
3^{ 5 x - 10} = 1
25^{x + 1} = 125^{ 3 x - 4}
\dfrac{1}{3^{x - 3}} = \sqrt[3]{9}
\left(\dfrac{1}{9}\right)^{x + 5} = 81
\left(\dfrac{1}{8}\right)^{x - 3} = 16^{ 4 x - 3}
\dfrac{25^{y}}{5^{4 - y}} = \sqrt{125}
8^{x + 5} = \dfrac{1}{32 \sqrt{2}}
30 \times 2^{x - 6} = 15
2^{x} \times 2^{x + 3} = 32
3^{x} \times 9^{x - k} = 27
a^{x + 1} = a^{3} \sqrt{a}
3^{x^{2} - 3 x} = 81
27 \left(2^{x}\right) = 6^{x}
Solve for y in each of the following logarithmic equations:
\log_{7} y = 5
\log_{16} y = \dfrac{1}{2}
\log_{y} 8 = 3
2\log_{2} y + 3\log_{2} y = 15
\log_{6} y^2 = 4
\log_{2} y^{4} + \log_{2} y = 10
Given the following functions, find its inverse.
f(x) = 2^{x-3} + 4
f(x) = 3^{x-2} + 5
f(x) = 5^{x+4}
f(x) = 4^{x+1}
f(x) = 6^{x} + 8
f(x) = 2^{x} - 9
Given the following functions , find its inverse.
f(x) =\text{log}_{2}(x - 5) + 3
f(x) = \text{log}_{3}(x - 4) + 2
f(x) = \text{log}_{4}(x - 6) + 1
f(x) = \text{log}_{5}(x - 7) + 4
f(x) = \text{log}_{6}(x - 8) + 5
f(x) = \text{log}_{7}(x - 9) + 6
Using logarithms, rewrite the exponential equation 5^{2x} = 25 in a different form.
Using the properties of exponents, simplify the expression (x^{2}y^{3})^{4}.
Using the properties of logarithms, simplify the expression \text{log}_{b}(x^{3}y^{2}).
Solve the inequality involving logarithms: \text{log}_{3}(x) \geq 2.
Consider the function f(x) = 2^{x-3} + 4. Show how a small change in x affects the value of f(x). What does this tell you about the rate of change of this function?
The growth of a certain bacteria population can be modeled by the exponential function f(t) = 200 \cdot 2^{\frac{t}{3}}, where t is the time in hours. After how many hours will the bacteria population reach 800?
A certain city's population grew from 100\,000 to 125\,000 over 2 years. Assuming exponential growth, write a function to model the city's population, P, as a function of time, t, in years since the population was 100\,000.