Logarithms: Introduction to "The Relationship"

Logarithmic and exponential functions - Topics in precalculus Exponential functions Inverse relations Exponential and logarithmic equations Proofs of Logarithm Properties (with worked solutions & videos) OML Search In these lessons, we will look at the four properties of logarithms and their proofs. They are the product rule, quotient rule, power rule and change of base rule. You may also want to look at the lesson on how to use the logarithm properties. Working with Exponents and Logarithms What is an Exponent? What is a Logarithm? A Logarithm goes the other way. It asks the question "what exponent produced this?": 4.2 - Logarithmic Functions and Their Graphs Inverse of Exponential Functions We stated in the section on exponential functions, that exponential functions were one-to-one. One-to-one functions had the special property that they have inverses that are also functions.

CHANGING THE BASE OF A LOGARITHM Let a, b, and x be positive real numbers such that and (remember x must be greater than 0). Then can be converted to the base b by the formula Let's verify this with a few examples. Exponential Function Reference This is the Exponential Function: f(x) = ax a is any value greater than 0 Properties depend on value of "a" When a=1, the graph is a horizontal line at y=1 Apart from that there are two cases to look at: a between 0 and 1 Basic Log Rules / Expanding Log Expressions Basic Log Rules / Expanding Logarithmic Expressions (page 1 of 5) Sections: Basic log rules, Expanding, Simplifying, Trick questions, Change-of-Base formula You have learned various rules for manipulating and simplifying expressions with exponents, such as the rule that says that x3 × x5 equals x8 because you can add the exponents. There are similar rules for logarithms. Log Rules:

Lesson LOGARITHMS AND EXPONENTIAL AND LOGARITHMIC EQUATIONS This lesson covers an overview of LOGARITHMS AND EXPONENTIAL AND LOGARITHMIC EQUATIONS Logarithmic Equations and Exponential Equations are two sides of the same coin. Logarithms are used to solve exponential equations and exponents are used to solve logarithmic equations. For more information on exponents, see the lesson on EXPONENTS. if and only if This means that the log of x to the base b equals y if and only if the base b raised to the power of the exponent y = x.

The Change-of-Base Formula The Change-of-Base Formula (page 5 of 5) Sections: Basic log rules, Expanding, Simplifying, Trick questions, Change-of-Base formula There is one other log "rule", but it's more of a formula than a rule. You may have noticed that your calculator only has keys for figuring the values for the common (base-10) log and the natural (base-e) log, but no other bases. Some students try to get around this by "evaluating" something like "log3(6)" with the following keystrokes: Of course, they get the wrong answer, because the above actually calculates the value of "log10(3) × 6".

Review : Exponential and Logarithm Equations In this section we’ll take a look at solving equations with exponential functions or logarithms in them. We’ll start with equations that involve exponential functions. The main property that we’ll need for these equations is, Now that we’ve seen a couple of equations where the variable only appears in the exponent we need to see an example with variables both in the exponent and out of it. The next equation is a more complicated (looking at least…) example similar to the previous one.