PROBLEMS USING INVERSE PROPERTIES LOGARITHMS

Problem 1 :

Simplify 

10 ^{log 4}

Solution :

Given that, 10 ^{log 4}

f (g(x)) = b^log_b ^x = x

10 ^{log 4} = 4

(because base and base of logarithm at the power both are the same)

Problem 2 :

Simplify 

log₅ 25^x

Solution :

= log₅ 25^x

Writing 25 in exponential form, we get

5²

= log₅ (5²)^x

= log₅ (5^2x)

This matches with the property, g(f (x)) = log_b b^x = x

= 2x

Problem 3 :

Find the inverse of each function.

a) f (x) = 6^x

b) y = ln(x + 3)

Solution :

a) f (x) = 6^x

Let g(x) be the inverse of the given function.

Using inverse properties of logarithm, g(x) = log₆ x

b) y = ln(x + 3)

Let g(x) be the inverse of the given function.

e^y = (x + 3)

Solving for x,

e^y - 3 = x

Exchange x to y and y to x

y = e^x - 3

g(x) = e^x - 3

Problem 4 :

Skydivers use an instrument called an altimeter to track their altitude as they fall. The altimeter determines altitude by measuring air pressure. The altitude h (in meters) above sea level is related to the air pressure P (in pascals) by the function shown in the diagram. What is the altitude above sea level when the air pressure is 57,000 pascals?

Solution :

h = -8005 ln (p/101300)

When p = 57000

h = -8005 ln (57000/101300)

h = -8005 ln (0.562)

= -8005(-0.576)

= 4612.9

Approximately 4613 m.

Problem 5 :

The pH value for a substance measures how acidic or alkaline the substance is. It is given by the formula pH = −log[H⁺], where H⁺ is the hydrogen ion concentration (in moles per liter). Find the pH of each substance.

a. baking soda: [H⁺] = 10⁻⁸ moles per liter

b. vinegar: [H⁺] = 10⁻³ moles per liter

Solution :

pH = −log[H⁺]

a) pH substance in baking soda :

Applying [H⁺] = 10⁻⁸

pH = −log[10⁻⁸]

= - (-8)

= 8

a) pH substance in vinegar :

Applying [H⁺] = 10⁻³

pH = −log[10⁻³]

= - (-3)

= 3

Problem 6 :

The energy magnitude M of an earthquake can be modeled by

M = (2/3) log E − 9.9

where E is the amount of energy released (in ergs).

a. In 2011, a powerful earthquake in Japan, caused by the slippage of two tectonic plates along a fault, released 2.24 × 10²⁸ ergs. What was the energy magnitude of the earthquake?

b. Find the inverse of the given function. Describe what the inverse represents.

Solution :

M = (2/3) log E − 9.9

a) E is the amount of energy released 

M = (2/3) log (2.24 × 10²⁸) − 9.9

Using the properties of logarithm, 

= (2/3) [log (2.24) + (log 10²⁸)] − 9.9

= (2/3) [0.350 + 28] − 9.9

= (2/3) [28.350] − 9.9

= 18.9 - 9.9

= 9

So, the energy of magnitude is 9 

b) Finding inverse :

M + 9.9 = (2/3) log E

log E = (3/2) M + 9.9

E = 10^{(3/2) M + 9.9}

Problem 7 :

The wind speed s (in miles per hour) near the center of a tornado can be modeled by s = 93 log d + 65, where d is the distance (in miles) that the tornado travels.

a. In 1925, a tornado traveled 220 miles through three states. Estimate the wind speed near the center of the tornado.

b. Find the inverse of the given function. Describe what the inverse represents

Solution :

s = 93 log d + 65

Here d is the distance (in miles) that the tornado travels.

s is the wind speed s (in miles per hour)

a) When d = 220 miles, s = ?

s = 93 log 220 + 65

s = 93(2.342) + 65

= 217.806 + 65

= 282.806

So, the required speed 283 miles per hour.

b) 

s = 93 log d + 65

s - 65 = 93 log d

(s - 65)/93 = log d

d = 10^{(s - 65)/93}

Problem 8 :

The function represents distance traveled with the speed of s miles per hour.

The lengthℓ (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 :

ℓ = 266 − 219e^−0.05t

When l = 175, t = ?

Finding inverse function :

219e^−0.05t = 266 - l

e^−0.05t = (266 - l)/219

e^0.05t = 219/(266 - l)

Applying l = 175, we get

e^0.05t = 219/(266 - 175)

e^0.05t = 219/91

e^0.05t = 2.40

0.05t = ln(2.40)

0.05t = 0.87

t = 0.87/0.05

t = 17.4

So, the required age is 18 years.