USE PROPERTIES OF LOGARITHMS TO REWRITE THE EXPRESSION WORKSHEET

Write as a single logarithm in the form log k:

Problem 1 :

log 6 + log 5

Solution

Problem 2 :

log 10 - log 2

Solution

Problem 3 :

2 log 2 + log 3

Solution

Problem 4 :

log 5 - 2 log 2

Solution

Problem 5 :

1/2 log 4 - log 2

Solution

Problem 6 :

log 2 + log 3 + log 5

Solution

Problem 7 :

log 20 + log (0.2)

Solution

Problem 8 :

- log 2 - log 3

Solution

Problem 9 :

3 log (1/8)

Solution

Problem 10 :

4 log 2 + 3 log 5

Solution

Problem 11 :

6 log 2 - 3 log 5

Solution

Problem 12 :

1 + log 2

Solution

Problem 13 :

1 - log 2

Solution

Problem 14 :

2 - log 5

Solution

Problem 15 :

3 + log 2 + log 7

Solution

Problem 16 :

Find the value of log₂ 4 log₄ 6 log₆ 8

Solution

Problem 17 :

log₃ 243 = 2x + 1

Solution

Problem 18 :

log₆ 36 = 5x + 3

Solution

Problem 19 :

e^-2x+1 = 13 

Solution

Problem 20 :

log₅(x² + x + 4) = 2

Solution

Solve the following logarithmic equations :

Problem 1 :

log₇ 3 + log₇ x = log₇ 32

Solution

Problem 2 :

2 log₆ 4x = 0

Solution

Problem 3 :

log₂ x + log₂ (x - 3) = 2

Solution

Problem 4 :

log₂ (x + 5) - log₂ (x - 2) = 3

Solution

Problem 5 :

4 ln (2x + 3) = 11

Solution

Problem 6 :

log x - log 6 = 2 log 4

Solution

Problem 7 :

log 2x = 1.5

Solution

Problem 8 :

log₂ 2x = -0.65

Solution

Problem 9 :

1/3 log₂ x + 5 = 7

Solution

Problem 10 :

4 log₅ (x + 1) = 4.8

Solution

Problem 11 :

log₂ x + log₂ 3 = 3

Solution

Problem 12 :

2 log₄ x - log₄ (x - 1) = 1

Solution

Solve the equation. Check for extraneous solutions.

Problem 13 :

ln (7x − 4) = ln (2x + 11)

Solution

Problem 14 :

log₂(x − 6) = 5

Solution

Problem 15 :

log 5x + log (x − 1) = 2

Solution

Problem 16 :

log₄(x + 12) + log₄ x = 3

Solution

Solve the following logarithmic equations.

Problem 1 :

ln x = -3

Solution

Problem 2 :

log(3x - 2) = 2

Solution

Problem 3 :

2 logx = log2 + log(3x - 4)

Solution

Problem 4 :

log x + log(x - 1) = log(4x)

Solution

Problem 5 :

log₃(x + 25) - log₃(x - 1) = 3

Solution

Problem 6 :

log₉(x - 5) + log₉(x + 3) = 1

Solution

Problem 7 :

log x + log(x - 3) = 1

Solution

Problem 8 :

log₂(x - 2) + log₂(x + 1) = 2

Solution

Problem 9 :

Given that 

2 log₃(x - 5) - log₃(2x - 13) = 1

Show that x² - 16x + 64 = 0 and solve for x.

Solution

Problem 10 :

a) Find the positive value of x such that 

log _x64 = 2

b) Solve for x 

log₂(11 - 6x) = 2log₂(x - 1) + 3

Solution

Problem 11 :

Given that a and b are positive constants, solve the simultaneous equations

a = 3b

log₃ a + log₃ b = 2

Give your answers as exact numbers.

Solution

Use the One-to-One Property to solve the equation for x.

Problem 1 :

log₂(x + 1) = log₂ 4

Solution

Problem 2 :

log₂(x - 3) = log₂ 9

Solution

Problem 3 :

log(2x + 1) = log 15

Solution

Problem 4 :

log(5x + 3) = log 12

Solution

Problem 5 :

ln(x + 2) = ln 6

Solution

Problem 6 :

ln(x - 4) = ln 2

Solution

Problem 7 :

ln(x² - 2) = ln 23

Solution

Problem 8 :

ln(x² - x) = ln 6

Solution

Problem 9 :

A population of 30 mice is expected to double each year. The number p of mice in the population each year is given by p = 30(2ⁿ). In how many years will there be 960 mice in the population?

Solution

Problem 10 :

Approximate the solution of each equation using the graph

1 − 5^{5 − x} = −9

Solution

Problem 11 :

Approximate the solution of each equation using the graph

log₂5x = 2

Solution

Problem 12 :

The apparent magnitude of a star is a measure of the brightness of the star as it appears to observers on Earth. The apparent magnitude M of the dimmest star that can be seen with a telescope is

M = 5 log D + 2

where D is the diameter (in millimeters) of the telescope’s objective lens. What is the diameter of the objective lens of a telescope that can reveal stars with a magnitude of 12?

Solution

Problem 13 :

A biologist can estimate the age of an African elephant by measuring the length of its footprint and using the equation

ℓ = 45 − 25.7e^−0.09a

where ℓ is the length (in centimeters) of the footprint and a is the age (in years).

a. Rewrite the equation, solving for a in terms of ℓ.

b. Use the equation in part (a) to find the ages of the elephants whose footprints are shown.

Solution