Chapter Contents

• Exponential & Logarithmic Functions
• 1. Definitions: Exponential and Logarithmic Functions
• 2. Graphs of Exponential and Logarithmic Equations
• 3. Logarithm Laws
• 4. Logarithms to Base 10
• 5. Natural Logarithms (base e)
• Dow Jones Industrial Average
• Calculating the value of e
• 6. Exponential and Logarithmic Equations
• World Population Live
• 7. Graphs on Logarithmic and Semilogarithmic Paper
• Exponential & Logarithmic Functions Problem Solver

• Comments, Questions?

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3. The Logarithm Laws

by M. Bourne

Since a logarithm is simply an exponent which is just being written down on the line, we expect the logarithm laws to work the same as the rules for exponents, and luckily, they do.

Exponents	Logarithms
`b^m × b^n = b^(m+n)`	` log_b xy = log_b x + log_b y`
`b^m ÷ b^n = b^(m-n)`	`log_b (x/y) = log_b x − log_b y`
`(b^m)^n = b^(mn)`	`log_b (x^n) = n log_b x`
`b^1 = b`	`log_b (b) = 1`
`b^0 = 1`	`log_b (1) = 0`

Note: On our calculators, "log" (without any base) is taken to mean "log base 10". So, for example "log 7" means "log₁₀7".

Examples

1. Expand

log 7x

as the sum of 2 logarithms.

Answer

2. Using your calculator, show that

`log (20/5) = log\ 20 − log\ 5`.

Answer

3. Express as a multiple of logarithms: log x⁵.

Answer

Note 1: Each of the following is equal to 1:

log₆ 6 = log₁₀ 10 = log_x x = log_a a = 1

The equivalent statements, using ordinary exponents, are as follows:

6¹ = 6

10¹ = 10

x¹ = x

a¹ = a

Note 2: All of the following are equivalent to `0`:

log₇ 1 = log₁₀ 1 = log_e1 = log_x 1 = 0

The equivalent statments in exponential form are:

7⁰ = 1

10⁰ = 1

e⁰ = 1

x⁰ = 1

Exercises

1. Express as a sum, difference, or multiple of logarithms:

`log_3((root(3)y)/8)`

Answer

2. Express

2 log_e 2 + 3 log_e n

as the logarithm of a single quantity.

Answer

Note: The logarithm to base e is a very important logarithm. You will meet it first in Natural Logs (Base e) and will see it throughout the calculus chapters later.

3. Determine the exact value of:

`log_3root(4)27`

Answer

4. Solve for y in terms of x:

log₂x + log₂y = 1

Answer