The Basics

The Logarithm: a secret dance that the trees do when we are not looking.

Actually, the logarithm is far more complicated than that. You have learned about logarithms before, but just to catch you up, here are the basics. Logarithms are inverses of a number to a power. The general equation for a logarithmic function is:

y = log_bx

This actually means x is the answer (or the logarithm) to b raised to the y power or:

x = b^y

b is the base, so when we are talking about logarithms, b is the base of the exponent.

Sample Problem

Show 2 × 2 × 2 × 2 × 2 × 2 in logarithmic and exponential form.

2 × 2 × 2 × 2 × 2 × 2 = 64
log₂64 = 6
2⁶ = 64

Sample Problem

Okay, let's try to evaluate a logarithm:

Evaluate log₁₀1000. So we ask ourselves 10 to what power is 1000?

10³ = 1000
log₁₀1000 = 3

In words, we would say the logarithm is 3. What we have really been talking about here is an inverse. The inverse of the logarithm of base 10 is 10 to the logarithm. We will talk about inverses of the logarithm graphs in the next section but it is helpful to understand what log really means.

Sample Problem

Evaluate the logarithm:

log₅. So we ask ourselves 5 to what power is ? Remember the square root is a number to the  power.

Sample Problem

Evaluate this:

3^log₃7. 3 is the base, and 3 is the base in the logarithm, so because they are inverses, the answer is 7! How easy was that?

Sample Problem

Evaluate:

log₁₀10².

Again, we are applying the inverses here since the log with base 10 is the inverse of an exponent with base 10. The answer is 2. Badabing-badabang.

Sample Problem

Evaluate log 235.

Wait, where is the base? If there is not base, that means it is a base of 10. Logarithm with base 10 is called a common logarithm. It would be hard to figure out 10 to what power is 235. It will end up being a decimal. We will have to evaluate this logarithm on your calculator!

We are not sure what calculator you have but if you have a scientific calculator, usually the log function thinks you are in base 10. Find the log button on your calculator and type log 235. The answer rounded to the hundredth is: 2.37.

If you want to check this, take 2.37¹⁰ which is 234.423. This is not 35 because we rounded. It would be more accurate if we checked this way: 10^2.371067862 which is very close to 235.

Why even use logarithmic functions? We have used these for centuries before computers came along and it was to help us multiply large numbers or take the powers of numbers. For instance, instead a calculator, there were tables of logarithms. We don't need old tables of logarithms or slide rules (old-fashioned calculators for old people), we can use calculators. Goodbye 18th century.

There are a few properties you have learned before in Algebra 2, but as a review, here they are:

Properties of Logarithms

Product Rule: log_xab = log_xa + log_xb
Quotient Rule:
Power Rule: log_xa^b = blog_xa

Sample Problem

Rewrite log 3ab³ as a sum of logarithms with no exponents:

log3ab³ = log(3) + log(a) + log(b³)
= log(3) + log(a) + 3log(b)