6. Exponential and Logarithmic Equations

by M. Bourne

Solving Exponential Equations using Logarithms

The logarithm laws that we met earlier are particularly useful for solving equations that involve exponents.

Example 1:

World Population

Don't miss the interactive Flash applet on world population below.

Go to World population.

Solve the equation 3^x = 12.7.

Answer

Example 2:

Two populations of bacteria are growing at different rates. Their populations at time t are given by 5^t+2 and e^2t respectively. At what time are the populations the same?

Answer

Exercises

1. Solve 5^x = 0.3

Answer

2. Solve 3 log(2x − 1) = 1

Answer

3. Solve for x: log₂ x + log₂ 7 = log₂ 21

Answer

4. Solve for x:

Answer

5. [Reader's question.]

I have the following formula:

S(n) = 5500 log n + 15000 (Using base 10)

If I know S(n) = 40 million, How do I solve it?

Answer

Application - World population growth

The population of the earth is growing at approximately 1.3% per year. The population at the beginning of 2000 was just over 6 billion. After how many more years will the population double to 12 billion?

Answer: We need an expression for the population at time t.

After one year, the population will be 1.3% higher than in 2000. (1.3% = 0.013)

Population after 1 year: 6 billion × 1.013.

Population after 2 years: 6 billion × (1.013)².

Population after 3 years: 6 billion × (1.013)³.

So our population, P, after t years, is given by:

P(t) = 6 billion × (1.013)^t

[In general, for any population growth,

P(t) = P₀(1 + r)^t

where P₀ is the population at time t = 0, r is the rate of growth per time period and t is the time.]

We are asked to find when the population doubles, so we need to solve:

12 000 000 000 = 6 000 000 000 × (1.013)^t

This gives 2 = (1.013)^t

Taking logarithms of both sides, we have:

log 2 = log (1.013)^t

Using the third log law, we have:

log 2 = t log (1.013)

So

So it will take only about 54 years to double the world's population, if it continues to grow at the current rate.

When the world population is 12 billion, the net number of people in the world will be increasing at the rate of about 5 per second, if the growth rate is still 1.3%. Currently, there are about 2.6 new people per second. However, the rate of growth is expected to drop considerably to about 0.5% within 50 years.

In 2001, the population of India passed one billion, making it the second country after China to reach that scary milestone.

Loading Flash movie.

Flash Interactive - World Population

Go to the interactive World Population, which has comparisons between present, past and future population growth.

In the following graph, we see that the population will be 10 billion by about 2030! It's got to stop! Think of our water quality, air pollution, global warming, social cohesion and lack of food. Surely this is one of the most important graphs in all of mathematics.

But I digress.

We are, of course, talking American English, here. The British billion has 12 zeroes (Well, even they have recently adopted the 9 zeroes billion...).