Solve Exponential and Logarithm Problems

Problems on Exponential and logarithm, some of the most important concepts in mathematics, are presented along with detailed solutions.

Problem 1:
The population P of a city increases according to the formula

P = 5000 e a t

where t is in years and t = 0 corresponds to 1980. In 1990, the population was 10000. Find the value of the constant a and approximate your answer to 3 decimal places.

Detailed Solution.

Problem 2:
The populations P1 and P2 of two cities are given by the formulas

P1 = 10000 e k t

P2 = 20000 e 0.01 t

where k is a constant and t is the time in years with t = 0 corresponding to the year 2000. Find constant k so that the two populations are equal in the year 2020 and approximate your answer to 3 decimal places.

Detailed Solution.

Problem 3:
The level of sound D in decibels is defined as follows

D = 10 log( I / 10 -16 )

where I is the sound intensity in watts per centimeters squared. Determine the level in decibels of a sound with intensity I = 10 -8 watts/cm 2.

Detailed Solution.

Problem 4:
Two sounds of intensities I1 and I2 have decibel levels of 60 and 80 respectively. Use the formula for decibel level given in problem 3 to determine the ratio of the intensities I2 / I1?

Detailed Solution.

Problem 5:
The spread of a virus through a city is modeled by the function

N = 15000 / [ 1 + 100 e -0.5 t ]

where N is the number of people infected by the virus after t days. How many days it takes for 2000 people of this city to be infected with the virus? (approximate your answer to 3 decimal places).

Detailed Solution.

Problem 6:
The amount of a radioactive material decays according to the formula

A(t) = A o e -k t

where A o is the initial amount, k is a positive constant and t is the time in days. Find a formula for the half life of the material.

Detailed Solution.

Problem 7:
A logistic equation has the form

y = a / [1 + b e -k t]

where a, b and k are constants. Solve the above equation for t.

Detailed Solution.

Problem 8:
The voltage V(t) across an electrical component is given by

V(t) = V o e -a t

where a is a positive constant depending on the values of the electronic components included in the electrical circuit and V o is the initial voltage.
Find t for which V(t) is equal to 50% of V o.

Detailed Solution.

Problem 9:
The signal ratio in decibels of an electronic system is given by the formula

10 Log (P o / Pi)

where P o is the output power and Pi is the input power of the system. Find the input power Pi if the output power is equal to 10 mw and the signal ratio in decibels is equal to 10 decibels.

Detailed Solution.

Problem 10:
The current i flowing in an RL circuit is given by

i = (E/R)[1 - e -Rt/L]

where E is the voltage applied to the circuit, R is the resistance and L is the inductance. Express i in terms of E and R when t = L/R.

Detailed Solution.

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