Properties of Logarithms Section 3.3

Properties of Logarithms What logs can we find using our calculators? ◦ Common logarithm ◦ Natural logarithm Although these are the two most frequently used logarithms, you may need to evaluate other logs at times For these instances, we have a change-of-base formula

Properties of Logarithms Change-of-Base Formula Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1. Then can be converted to a different base as follows: Base b

Properties of Logarithms Change-of-Base Formula Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1. Then can be converted to a different base as follows: Base 10

Properties of Logarithms Change-of-Base Formula Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1. Then can be converted to a different base as follows: Base e

Properties of Logarithms Evaluate the following logarithm: → 4 raised to what power equals 30? Since we don’t know the answer to this, we would want to use the change-of-base formula

Properties of Logarithms Evaluate the following logarithm using the natural log function.

Properties of Logarithms Evaluate the following logarithms using the common log and the natural log. a) b)

Properties of Logarithms What is a logarithm? Therefore, logarithms should have properties that are similar to those of exponents

Properties of Logarithms For example, evaluate the following: a) b) c)

Properties of Logarithms Just like we have properties for exponents, we have properties for logarithms. These properties are true for logs with base a, the common logs, and the natural logs

Properties of Logarithms Let a be a positive number such that a ≠ 1, and let n be a real number. If u and v are positive real numbers, the following properties are true.

Properties of Logarithms Use the properties to rewrite the following logarithm: From property 1, we can rewrite this as the following:

Properties of Logarithms Use the properties to rewrite the following logarithm: From property 2, we can rewrite this as the following:

Properties of Logarithms Use the properties to rewrite the following logarithm: From property 3, we can rewrite this as the following:

Properties of Logarithms Section 3.3

Properties of Logarithms Yesterday: a) Change-of-Base Formula b) 3 Properties

Properties of Logarithms Today we are going to continue working with the three properties covered yesterday.

Properties of Logarithms These properties can be used to rewrite log expressions in simpler terms We can take complicated products, quotients, and exponentials and convert them to sums, differences, and products

Properties of Logarithms Expand the following log expression: Start by applying property 1 to separate the product:

Properties of Logarithms Expand the following log expression: Apply property 3 to eliminate the exponent

Properties of Logarithms Expand the following expression: Start by applying property 1 to separate the product:

Properties of Logarithms Expand the following expression: Eliminate the exponents

Properties of Logarithms Rewrite the following logarithm: For problems involving square roots, begin by converting the square root to a power

Properties of Logarithms Apply property 1 to get rid of the quotient:

Properties of Logarithms Apply property 3 to get rid of the exponent

Properties of Logarithms Rewrite the following logarithmic expressions:

Properties of Logarithms Expand the following expression:

Properties of Logarithms

Section 3.3

Properties of Logarithms So far in this section, we have: a) Change-of-Base Formula b) 3 Properties c) Expanded expressions Today we are going to do the exact opposite d) Condense expressions

Properties of Logarithms When we were expanding, what order did we typically apply the properties in? ◦ Property 1 or Property 2 ◦ End with Property 3 When we condense, we use the opposite order ◦ Property 3 ◦ Property 1 or Property 2

Properties of Logarithms The most common error: Log x – Log y When you condense, you are condensing the expression down to one log function

Properties of Logarithms Condense the following expression: Start by applying property 3, then move on to properties 1 and 2

Properties of Logarithms Is this expression simplified to one log function?

Properties of Logarithms Condense the following expression:

Properties of Logarithms