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2.13 Exponential and logarithmic equations and inequalities

Worksheet
What do you remember?
1

State the properties of exponents.

2

State the properties of logarithms.

3

Explain how to solve an exponential equation.

4

What is the general form of an exponential function and its transformations?

5

How do you find the inverse of an exponential function?

Let's practice
6

Solve the following exponential equations:

a

\left(\sqrt{6}\right)^{y} = 36

b

\left(\sqrt{2}\right)^{k} = 0.5

c

9^{y} = 27

d

3^{ 5 x - 10} = 1

e

25^{x + 1} = 125^{ 3 x - 4}

f

\dfrac{1}{3^{x - 3}} = \sqrt[3]{9}

g

\left(\dfrac{1}{9}\right)^{x + 5} = 81

h

\left(\dfrac{1}{8}\right)^{x - 3} = 16^{ 4 x - 3}

i

\dfrac{25^{y}}{5^{4 - y}} = \sqrt{125}

j

8^{x + 5} = \dfrac{1}{32 \sqrt{2}}

k

30 \times 2^{x - 6} = 15

l

2^{x} \times 2^{x + 3} = 32

m

3^{x} \times 9^{x - k} = 27

n
a^{x-1} = a^4
o

a^{x + 1} = a^{3} \sqrt{a}

p

3^{x^{2} - 3 x} = 81

q

27 \left(2^{x}\right) = 6^{x}

r
3^{x} \times 3^{ n x} = 81
7

Solve for y in each of the following logarithmic equations:

a

\log_{7} y = 5

b

\log_{16} y = \dfrac{1}{2}

c

\log_{y} 8 = 3

d

2\log_{2} y + 3\log_{2} y = 15

e

\log_{6} y^2 = 4

f

\log_{2} y^{4} + \log_{2} y = 10

8

Given the following functions, find its inverse.

a

f(x) = 2^{x-3} + 4

b

f(x) = 3^{x-2} + 5

c

f(x) = 5^{x+4}

d

f(x) = 4^{x+1}

e

f(x) = 6^{x} + 8

f

f(x) = 2^{x} - 9

9

Given the following functions , find its inverse.

a

f(x) =\text{log}_{2}(x - 5) + 3

b

f(x) = \text{log}_{3}(x - 4) + 2

c

f(x) = \text{log}_{4}(x - 6) + 1

d

f(x) = \text{log}_{5}(x - 7) + 4

e

f(x) = \text{log}_{6}(x - 8) + 5

f

f(x) = \text{log}_{7}(x - 9) + 6

10

Using logarithms, rewrite the exponential equation 5^{2x} = 25 in a different form.

11

Using the properties of exponents, simplify the expression (x^{2}y^{3})^{4}.

12

Using the properties of logarithms, simplify the expression \text{log}_{b}(x^{3}y^{2}).

13

Solve the inequality involving logarithms: \text{log}_{3}(x) \geq 2.

Let's extend our thinking
14

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?

15

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?

16

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.

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Outcomes

2.13.A

Solve exponential and logarithmic equations and inequalities.

2.13.B

Construct the inverse function for exponential and logarithmic functions.

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