Logarithms

A logarithm is the inverse of taking something to an exponential power. When something is expressed as log₅125, in a sense it’s asking what exponent would 5 need to be taken to to get 125. This is why log₅125 = 3.That’s why the first core skill of logarithms is to be able to rewrite logs in exponential form.

Rewrite Exponentially

If given log_ax = b, be able to rewrite this formula as a^b= x. This will make any question asking you to solve for a variable in the original equation much more intuitive and easy to answer.

Log Operations

In the same way that exponents have certain rules for fundamental math operators, logarithms have their own rules for adding, subtracting, multiplying, and dividing. Just like with exponents, these rules generally rely on having the same base number.

Add

Log_a(X•Y) = Log_aX  + Log_aY The log of any given number is equal to the sum of the logs of its factors. Or read the other way, two logs added together is equivalent to the log of their product

Subtract

Loga(XY)=LogaX –LogaY The log of any given fraction is equal to the log of the numerator minus the log of the denominator. Or read the other way, one log subtracted from another is equal to the ratio between those two numbers.

Special Skills and Facts

Moving Exponents

Since logs are related to exponents, exponents that are a part of a log can be moved around in a convenient way! Specifically, exponents on numbers inside of the log function can be moved in front of the log and turned into a linear coefficient.

For example log_a x², can be rewritten as 2 log_a x.

Natural Log

A normal logarithm uses the shorthand log, but there are other types of logarithms that are commonly used in math and other fields. THe most common one is the natural log, which is written as ln x. This still follows the same rule as any other log, but instead of being the log of an integer, natural log is log base e, where e is the special number known as

Log Base 10

In the same way that without a specific number on the radical is understood to mean the square root. A log  without a specified base is understood to mean log₁₀.

Log 1

Another great example of how log rules relate to exponent rules. Any number taken to the power of 0 = 1. So, Log 1 = 0, no matter what the Log base is.