Properties of Logarithms

Properties of Logarithms Properties are based off of the rules of exponents (since exponents = logs) The base of the logarithm can not be equal to 1 and the values must all be positive (no negatives in logs) 𝑏=1 𝑏>0

Product Rule logbMN = LogbM + logbN Ex: logbxy = logbx + logby Ex: log6 = log 2 + log 3 Ex: log39b = log39 + log3b

Quotient Rule Ex:

Power Rule Ex: 𝑙𝑛 𝑥 = 1 2 ln⁡(𝑥)

Pre-Req to Solving Equations: Simplifying, Expanding, and Condensing. Let’s try condensing first…

Let’s try some Write the following as a single logarithm.

Let’s try something more complicated . . . Condense the logs log 5 + log x – log 3 + 4log 5

Let’s try something more complicated . . . Condense the logs log 5 + log x – log 3 + 4log 5

Using Properties to Expand Logarithmic Expressions Use exponential notation Use the product rule Use the power rule

Now we’ll try expanding…

Expand

More Properties of Logarithms This one says if you have an equation, you can take the log of both sides and the equality still holds. This one says if you have an equation and each side has a log of the same base, you know the "stuff" you are taking the logs of are equal.