Mastering Logarithms: ACT Math

What are Logarithms?

Logarithms (logs) are mathematical functions that represent the exponent to which a base must be raised to produce a given number. In simpler terms, logarithms help us solve equations involving exponential quantities. They are particularly useful for expressing large or small numbers in a more manageable form.

For example:

logarithm base 10 of 100, written as log(100)
(When the base is 10, we don’t write the subscript base.)

That means “Ten to the WHAT is 100?”
So you can think “Ten squared is 100.”

So:
log(100) is the same as 2.
log(100) = 2.

This tells us that 10 raised to the power of 2 equals 100 or 10^2=100

Similarly:

logarithm base 2 of 8, written as log₂(8)

That means “Two to the WHAT is 8?”
So you can think “Two cubed is 8.”

We say “Log Base 2 of 8 is 3.” or log₂(8) = 3

This means that 2 raised to the power of 3 equals 8 or 2 to the 3rd power is 8.

In my sessions we say: LOG IS EXPONENT to help remember what goes where.

Properties of Logarithms:

Logarithms possess several essential properties that facilitate their manipulation and calculation. These properties include:

1. Product Rule: log(a * b) = log(a) + log(b)

2. Quotient Rule: log(a / b) = log(a) - log(b)

3. Power Rule: log(a^b) = b * log(a)

These rules allow us to simplify logarithmic expressions, break them apart, and solve equations more efficiently.

Solving Logarithmic Equations:

When faced with logarithmic equations, the key is to isolate the logarithmic term and apply the appropriate properties to solve for the variable. Let's consider an example:

Given the equation log₃(x) = 2, we want to find the value of x. Using the definition of a logarithm, we rewrite the equation as 3^2 = x. Thus, x = 9.

Logarithmic Functions and Graphs:

Logarithmic functions, such as y = logₐ(x), exhibit distinct characteristics when graphed. They feature vertical asymptotes, domain restrictions, and specific behavior based on the base of the logarithm.

For instance, the graph of y = log(x) has a vertical asymptote at x = 0, and it increases slowly for x > 0. Conversely, the graph of y = logₐ(x), where a > 1, increases more rapidly.

Applications to ACT Math:

In the ACT Math section, logarithms often appear in problems involving exponential growth or decay, scientific notation, and complex equations. Mastering logarithms can significantly enhance your problem-solving abilities and increase your score.