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• Practice

a ln 𝑥

2

3

4

b log 𝑥

2

3

4

3

Use laws of logarithms to expand the following:

Exercise 2

Use the laws of logarithms to evaluate log

2

80 – log

2

Exercise 3

Solve each of the following equations:

a 2

3𝑥− 2

= 4 b 𝑥 − 𝑥𝑒

5𝑥− 3

c ln 3𝑥 − 10 = 2 d ln𝑥 − ln 2𝑥 − 2 = 3

Exercise 4

Find the inverse of each of the following functions

a 𝑦 = 5 − 7𝑥 b 𝑦 = 5𝑥 − 7 𝑐^ 𝑦^ =^

a ln 𝑥

2

3

4

b log 𝑥

2

3

4

3

Use laws of logarithms to expand the following:

Solution

a ln 𝑥

2

3

4

= ln 𝑥

2

• ln 𝑦

3

• ln 𝑧

4

= 2ln 𝑥 + 3 ln 𝑦 + 4 ln 𝑧

b log 𝑥

2

3

4

3

= log 𝑥

2

• log 𝑦

3

• log

4

3

= log 𝑥

2

• log 𝑦

3

• log(𝑥 + 2𝑦)

3 / 4

= 2log 𝑥 + 3 log 𝑦 +

log(𝑥 + 2𝑦)

Use the laws of logarithms to evaluate log

2

80 – log

2

Solution

log

2

80 – log

2

5 = log

2

= log

2

= log

2

4

Solve each of the following equations:

a 2

3𝑥− 2

= 4 b 𝑥 − 𝑥𝑒

5𝑥− 3

c ln 3𝑥 − 10 = 2 d ln𝑥 − ln 2𝑥 − 2 = 3

Solution

Take the natural logarithm of both sides

ln 𝑒

5𝑥− 3

= ln 1 

b 𝑥 − 𝑥𝑒

5𝑥− 3

5𝑥− 3

5𝑥− 3

= 0 So,^ x^ = 0 is a solution.

Look at  1 − 𝑒

5𝑥− 3

5𝑥− 3

So, the solution set is 1 ,

5𝑥 − 3 ln 𝑒 = ln (^1) 

Solve each of the following equations:

a 2

3𝑥− 2

= 4 b 𝑥 − 𝑥𝑒

5𝑥− 3

c ln 3𝑥 − 10 = 2 d ln𝑥 − ln 2𝑥 − 2 = 3

Solution

c ln 3𝑥 − 10 = 2 Exponentiate both sides

ln 3𝑥− 10

2

 (^) 3𝑥 − 10 = 𝑒

2



2

2

Find the inverse of each of the following functions

Solution

a 𝑦 = 5 − 7𝑥

Interchange x and y to get (^) x = 5 − 7𝑦

Solve for y

x = 5 − 7 𝑦

 𝑦^ =^

Therefore, the inverse of y = f ( x ) = 5 − 7 x is:

− 1

a 𝑦 = 5 − 7𝑥 b 𝑦 = 5𝑥 − 7 𝑐 𝑦 =

Find the inverse of each of the following functions

Solution

Interchange x and y to get (^) x = 5𝑦 − 7

Solve for y

2

− 1

2

b 𝑦 = 5𝑥 − 7

x = 5𝑦 − 7  5𝑦 = 𝑥

2

2

Therefore, the inverse of y = 𝑓 𝑥 = 5𝑥 − 7 is:

a 𝑦 = 5 − 7𝑥 b 𝑦 = 5𝑥 − 7 𝑐 𝑦 =

0

5

Sign of x −^ + +

Sign of x − 5 −^ − +

Sign of x ( x − 5)

• − +

Determine the solution of the inequality from the last row of the sign table

Dom ( h ) = (− , 0)  (5, )

Find the domain of the function

4

2

Solution

We require that 𝑥

2

2

Exercise 5 ( Section 1.1, Exercise 35 )

Find the domain of the function 𝑓 𝑥 =

Solution

We have two denominators in f , each must be different from zero.

≠ 0 ^

≠ 0 ^

Therefore, (^) Dom ( f ) = R{− 2 , −1}

Exercise 6 ( Section 1.1, Exercise 36 )

Given the functions f ( x ) = x and g x = 2 − x.

Find the following functions and their domains.

( ) b g f

Solution :

( ) ( b g f )( x ) = g f ( ( x )) =

g x = 2 − x

Dom( f g ) =[0, 4].

2 − x  0  x  2 

x  4.

x 

2 − x 

x 0.

2 − x 0.

Therefore:

0  x  4