# 6. Exponential and Logarithmic Equations

by M. Bourne

## Solving Exponential Equations using Logarithms

### World Population

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

Go to World population.

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

### Example 1

Solve the equation `3^x= 12.7`.

### 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?

### Exercises

1. Solve `5^x= 0.3`

2. Solve `3\ log(2x − 1) = 1`.

3. Solve for *x*:

`log_2 x + log_2 7 = log_2 21`

4. Solve for *x*:

`3\ ln\ 2+ln(x-1)=ln\ 24`

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?

## 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?

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.

### World population

Current world population is approximately:

**Loading...**

#### Interactive applet - World Population

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

### Predicting world population

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...).

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