SOLVING LOGARITHMIC EQUATIONS WORKSHEET

Solve logarithmic equations :

1)  log₇ 3 + log₇ x = log₇ 32                   Solution

2)  2 log₆ 4x = 0                 Solution

3)  log₂ x + log₂ (x - 3) = 2                 Solution

4)  log₂ (x + 5) - log₂ (x - 2) = 3                 Solution

5)  4 ln (2x + 3) = 11                 Solution

6)  log x - log 6 = 2 log 4                 Solution

7)  log 2x = 1.5                 Solution

8)  log₂ 2x = -0.65                 Solution

9)  1/3 log₂ x + 5 = 7                 Solution

10)  4 log₅ (x + 1) = 4.8                 Solution

11)  log₂ x + log₂ 3 = 3                 Solution

12)  2 log₄ x - log₄ (x - 1) = 1                 Solution

13) ln (7x − 4) = ln (2x + 11)          Solution

14) log₂(x − 6) = 5       Solution

15) log 5x + log (x − 1) = 2        Solution

16) og₄(x + 12) + log₄ x = 3       Solution

Find the value of y.

Solution

Problem 13 :

Describe the similarities and difference between in solving the equations 

4^{5x - 2} = 16 and log₄(10x + 6) = 1

Then solve the each equation

Solution

Problem 14 :

For a sound with intensity I (in watts per square meter) the loudness L(I) of the sound (in decibels) is given by the function

L(I) = 10 log (I/I₀)

Where I0 is the intensity of barely audible sound (about 10^-12 watts per square meter) An artist in a recording studio turns up the volume of a track so that the intensity of the sound doubles. By how many decibels does the loudness increase ?

Solution

Problem 15 :

The length ℓ (in centimeters) of a scalloped hammerhead shark can be modeled by the function

ℓ = 266 − 219e^−0.05t

where t is the age (in years) of the shark. How old is a shark that is 175 centimeters long?

Solution

Solve the following logarithmic equations.

Problem 1 :

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

Solution

Problem 2 :

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

Solution

Problem 3 :

log₃ (x + 25) - log₃ (x - 1) = 3

Solution

Problem 4 :

log₉ (x - 5) + log₉ (x + 3) = 1

Solution

Problem 5 :

log x + log (x - 3) = 1

Solution

Problem 6 :

log₂ (x - 2) + log₂ (x + 1) = 2

Solution

Problem 7 :

ln x = -3

Solution

Problem 8 :

log(3x - 2) = 2

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