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Introduction to Logarithms

Author: Sophia

what's covered
In this lesson, you will learn how to solve a logarithmic equation. Specifically, this lesson will cover:

Table of Contents

1. Relating Logarithms to Exponential Equations

There is an inverse relationship between logarithms and exponents. If we have the expression 3 to the power of x equals 9, we can gather that x equals 2, because 3 squared equals 9 (3 squared equals 9).

As a logarithmic expression, we can write this equivalently as log subscript 3 left parenthesis 9 right parenthesis equals 2. This reads, "the log, base 3, of 9 is 2." The expression tells us that the base number, 3, must be raised to the power of 2 in order to equal 9.

In general, we can write the relationship between logarithms and exponents as follows:

y equals b to the power of x Exponential equation
log subscript b open parentheses y close parentheses equals x Logarithmic equation

Notice that x and y switched as being isolated onto one side of the equals sign. This is characteristic of inverse relationships. Also, note that the base to the exponential is the base of the logarithm.

EXAMPLE

Rewrite the exponential equation 8 equals 2 to the power of x as a logarithmic equation

8 equals 2 to the power of x Exponential equation
log subscript 2 open parentheses 8 close parentheses equals x Logarithmic equation

hint
If you know your powers of 2, you may be able to gather that x equals 3 in this case.

term to know
Logarithm
The inverse of a power, the logarithm describes how many times a number should be multiplied by itself to result in another number.


2. Common Log and Natural Log

If you have a scientific calculator that can compute logarithms, there are likely two kinds of log buttons on your calculator: one that simply says "log" and another that says "ln." The first button, "log" is known as the common log, while the other, "ln," is referred to as the natural log.

They are both logarithms, but their difference is in their base. Common log operates under a base of 10. So if you ever see expressions such as log(42) or log(67), the base of the log is 10.

hint
Whenever a base is not explicitly written next to "log," it is assumed to be the common log, which is base 10.

The abbreviation "ln" comes from the Latin logarithmus naturali. The base of this logarithm is the mathematical constant "e". The constant "e", or Euler's constant, is approximately equal to 2.718282. If you have the natural log button (ln) on your calculator, definitely use it for the most accurate calculations. Otherwise, use the approximation 2.718282.

hint
ln, or natural log, operates in base e, which is approximately equal to 2.718282. ln(x) and loge(x) are the same expressions.


3. Evaluating Logarithmic Expressions

We can use the relationship between exponential equations and logarithmic equations to evaluate expressions by thinking about how many times we must multiply a given number by itself to result in another given number.

EXAMPLE

Evaluate log subscript 4 left parenthesis 64 right parenthesis.

log subscript 4 open parentheses 64 close parentheses Rewrite using exponents
4 to the power of x equals 64 4 cubed results in 64
4 cubed equals 64 Write the solution to expression
log subscript 4 open parentheses 64 close parentheses equals 3 Our solution

EXAMPLE

Evaluate log subscript 3 left parenthesis 243 right parenthesis.

log subscript 3 open parentheses 243 close parentheses Rewrite using exponents
3 to the power of x equals 243 3 raised to the 5th power is 243
3 to the power of 5 equals 243 Write the solution to expression
log subscript 3 open parentheses 243 close parentheses equals 5 Our solution

Notice how the bases are the same in both exponential and logarithmic form.

bold italic l bold italic o bold italic g subscript bold b open parentheses bold y close parentheses bold equals bold italic x bold italic y bold equals bold italic b to the power of bold x
log subscript 4 open parentheses 64 close parentheses equals 3 64 equals 4 cubed
log subscript 3 open parentheses 243 close parentheses equals 5 243 equals 3 to the power of 5

summary
In an exponential equation, a base number is raised to a variable power and represented as y equals b to the power of x. Relating logarithms to exponents, the input of the logarithmic function is the output of the exponential function, and the output of the logarithmic function is the input of the exponential function.

With logarithms, there are two types of log: common log and natural log. The common log is a logarithm with a base of 10. An expression that just has log and now base, like log(42), implies a base of 10. The natural log is a logarithm with a base e, where e is equal approximately to 2.718281. When evaluating logarithmic expression, you can rewrite using an exponential expression.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

Terms to Know
Logarithm

The inverse of a power, the logarithm describes how many times a number should be multiplied to result in another number.