Module 3 Lesson 12: Properties of Logarithms

Below are six properties of logarithms. These are tools that we can use

to help us to evaluate logarithms.

Property 1: A logarithm with any base and an argument of 1 will always

have a power of 0. log(1) = 0

log4(1) = log11(1) =

Property 2: If a logarithm’s base and argument match, the power will

equal 1. log(10) = 1

log4(4) = log13(13) =

Property 3: If a logarithm’s base and argument match, whatever

exponent the argument has will be the solution. log(10𝑟) = 𝑟

log4(45) = log6(6

8) =

Property 4: If a base is raised to a logarithm that has the same base, they

cancel each other out, releasing the argument. For 𝑥 > 0, 10log 𝑥 = 𝑥.

5log5 10 = 9log9 𝑥 =

Property 5: (The product property) For any positive real numbers x and y,

log(𝑥 ∙ 𝑦) = log(𝑥) + log(𝑦)

Property 6: (The power property) For any positive real numbers x and y

and any real number r, log(𝑥𝑟) = 𝑟 ∙ log(𝑥) .

When we need to estimate logarithms, we can use the properties of

logarithms to help. Use the approximation log(2) ≈ 0.3010 to

approximate the following.

log(20) log(0.2) log(24)

Apply properties of logarithms to rewrite the following expressions as a

single logarithm or number.

1

2log(25) + log(4)

1

3log(8) + log(16) 3 log(5) + log(.8)

Apply the properties of logarithms to expand the following expressions:

log(3𝑥2𝑦5) log(√𝑥7𝑦3)

Logarithmic Equations

We can use logarithms to solve equations by taking the log or raising

both sides of an equation to an equal power. Just beware, you cannot

take the log of a negative number or zero.

1010 = 100 10𝑥−1 =1

10𝑥+1 1002𝑥 = 103𝑥−1

10𝑥 = 27 10𝑥2+1 = 15 4𝑥 = 53