Summary
Highlights
A brief recap of the lesson covers the definition of logarithms and a summary of all the properties learned: product of arguments (summation), quotient of arguments (subtraction), argument to a power (multiplication), root of an argument (division), and change of base.
Logarithms of integers involve finding the exponent to which a base number must be raised to obtain another number, called the argument. For example, to solve log base 2 of 8, we find how many times 2 must be multiplied by itself to get 8, which is 3 (2*2*2 = 8).
This section demonstrates how to solve basic logarithms. If no base is explicitly given, the base is assumed to be 10. Examples include log base 10 of 100 (answer: 2), log base 5 of 125 (answer: 3), log base 3 of 81 (answer: 4), and log base 8 of 1 (answer: 0, as any number raised to the power of 0 is 1).
When there is a product of arguments within a logarithm, it can be separated into a sum of logarithms with the same base. For example, log base 2 of (4 * 8) can be written as log base 2 of 4 + log base 2 of 8, which simplifies to 2 + 3 = 5.
Similar to products, when there is a quotient (division) of arguments, it can be separated into a subtraction of logarithms with the same base. For example, log base 3 of (81 / 27) can be written as log base 3 of 81 - log base 3 of 27, which simplifies to 4 - 3 = 1.
If the argument of a logarithm is raised to a power, the exponent can be moved to multiply the entire logarithm. For instance, log base 4 of (64^5) becomes 5 * log base 4 of 64, which evaluates to 5 * 3 = 15.
When a logarithm has an argument under a root, the radical index can be moved to the denominator, dividing the logarithm's result. For example, log base 2 of the cube root of 64 is rewritten as log base 2 of 64 / 3, which is 6 / 3 = 2.
This activity involves simplifying an complex logarithmic expression using the properties discussed earlier. The expression log of the square root of (1000^3 * 100 / 10) is broken down step by step, applying product, quotient, root, and power properties, ultimately solving to 11/2.
This property allows changing the base of a logarithm. To change log base B of A to a new base C, it becomes (log base C of A) / (log base C of B). An example is changing log base 9 of 27 to base 3, resulting in (log base 3 of 27) / (log base 3 of 9), which simplifies to 3 / 2.
This section provides practice in changing the base of logarithms and then solving them. Examples include changing log base 16 of 64 to base 2, yielding 6/4 (or 3/2), and changing log base 100 of 10,000 to base 10, resulting in 4/2 (or 2).
A real-world problem about a bookshelf with books is presented to demonstrate the practical application of logarithms. The problem asks for the number of parts each section is divided into, given the total number of books, leading to solving log base 3 of 27, which is 3.