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Introduction

  • A logarithm is simply an alternative representation of exponents and proves useful in solving problems beyond the scope of exponentials alone. grasping the concept of logarithms is relatively straightforward. Essentially, understanding logarithms boils down to recognizing that a logarithmic equation is merely an alternative form of an exponential equation.
  • Logarithms and exponents are reciprocal concepts, as demonstrated in the following section. Initially introduced by mathematician John Napier to streamline calculations, logarithms swiftly gained traction among scientists, engineers, and others.

Here is the mathematical definition of logs.

Logs Definition

A logarithm is defined using an exponent.

  • b= a ⇔ logb a = x

Here, "log" stands for logarithm. The right side part of the arrow is read to be "Logarithm of a to the base b is equal to x".

A very simple way to remember this is "base stays as the base in both forms" and "base doesn't stay with the exponent in log form". Notice that 'b' is the base both on the left and right sides of the implies symbol and in the log form see that the base b and the exponent x don't stay on the same side of the equation.

Here,

  • a and b are two positive real numbers.
  • x is a real number.
  • a, which is inside the log is called the "argument".
  • b, which is at the bottom of the log is called the "base".

Logarithms | Mathematics for ACT

The above equation has two things to understand (from the symbol ⇔):

  • bx = a ⇒ logb a = x. This is called "exponential to log form"
  • logb a = x ⇒ bx = a . This is called "log to exponential form"

Here is a table to understand the conversions from one form to the other form.

Logarithms | Mathematics for ACT

Natural Logarithm and Common Logarithm

  • Examining the last two rows of the table provided, we encounter loge and log10, which hold particular significance and are bestowed with specific names within logarithmic principles.
  • The logarithm with base e, referred to as loge, assumes the title of "natural log." However, it is conventionally denoted by the symbol ln instead of loge.

For instance:

  • ex = 2 can be expressed as loge 2 = x or ln 2 = x.
  • ex = 7 can be articulated as loge 7 = x or ln 7 = x.

On the other hand, the logarithm with base 10, known as log10, earns the moniker of "common log." Typically, it is simply represented as log without any base specified, which inherently implies a base 10 logarithm.

For example:

  • 102 = 100 can be rendered as log10 100 = 2 or log 100 = 2.
  • 10-2 = 0.01 can be depicted as log10 0.01 = -2 or log 0.01 = -2.

It's noteworthy that in the examples provided, the base 10 is omitted from the notation as it is implicit.

Rules of Logs

The rules governing logarithms serve to simplify, expand, or consolidate multiple logarithms into a single expression. These rules, or properties, play a fundamental role in logarithmic operations. If one desires a deeper understanding of how these rules are derived, further exploration can be pursued.

Logarithms | Mathematics for ACT

Let's review each of these principles individually.

Log 1

  • The value of log 1 irrespective of the base is 0. Because from the properties of exponents, we know that, a0 = 1, for any 'a'. Converting this into log form, loga 1 = 0, for any 'a'. Obviously, when a = 10, log10 1 = 0 (or) simply log 1 = 0.
  • When we extend this to the natural logarithm, we have, since e= 1 ⇒ ln 1 = 0.

Loga a

Since a1 = a, for any 'a', converting this equation into log form, loga a = 1. Thus, the logarithm of any number to the same base is always 1. For example:

  • log2 2 = 1
  • log3 3 = 1
  • log 10 = 1
  • ln e = 1

Product Rule of Log

  • The logarithm of a product of two numbers equals the sum of the logarithms of the individual numbers. Symbolically:
  • loga (mn) = logm + loga n
  • This mirrors the product rule of exponents: xm ⋅ xn = x(m+n). For example:
  • log 6 = log (3 x 2) = log 3 + log 2
  • log (5x) = log 5 + log x

Quotient Rule of Log

Similarly, the logarithm of a quotient of two numbers equals the difference between the logarithms of the individual numbers:

  • loga (m/n) = loga m - loga n

This aligns with the quotient rule of exponents: xm / xn = x(m-n). For instance:

  • log 4 = log (8/2) = log 8 - log 2
  • log (x/2) = log x - log 2

Power Rule of Log

The exponent of the argument of a logarithm can be moved to the front of the logarithm:

  • loga (mn) = n loga m

This corresponds to the power of power rule of exponents: (xm)^n = x^(mn).

Change of Base Rule

This rule allows changing the base of a logarithm:

  • logb a = (log꜀ a) / (log꜀ b)

Alternatively, it can be expressed as logb a · log꜀ b = log꜀ a. This enables changing the base to any desired number, including 10:

  • log2 3 = (log 3) / (log 2)
  • log3 2 = (log 2) / (log 3)

Equality Rule of Logarithms

This rule is pivotal in solving logarithmic equations:

  • logb a = logb c ⇒ a = c

It simplifies equations involving logarithms by canceling out the logarithms from both sides.

Number Raised to Log Property

When a number is raised to a logarithm with the same base as the number, the result equals the argument of the logarithm:
aloga x = x
For example:
2log2 5 = 5
10log 6 = 6
eln 3 = 3

Negative Log Property

Negative logarithms are represented as −logb a. We can calculate this using the power rule of logarithms:
−logb a = logb a-1 = logb (1/a)

Thus,
−logb a = logb (1/a)
In other words, to convert a negative log into a positive log, we can take the reciprocal of the argument or the base:
−logb a = log1/b a

Condensing/Expanding Logarithms

  • Logarithms can be either compressed into a single log or expanded into multiple logs using the above rules. The key rules employed in this process include the product rule of logarithms, quotient rule of logarithms, and power rule of logarithms.

Expanding Logarithms:

The logarithm log (3x2y3) can be expanded as:
log (3x2y3)
= log (3) + log (x2) + log (y3) (By product rule)
= log 3 + 2 log x + 3 log y (By power rule)

Condensing Logarithms:
Conversely, the sum of logarithms log 3 + 2 log x + 3 log y can be condensed as:
log 3 + 2 log x + 3 log y
= log (3) + log (x2) + log (y3) (By power rule)
= log (3x2y3) (By product rule)

The document Logarithms | Mathematics for ACT is a part of the ACT Course Mathematics for ACT.
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FAQs on Logarithms - Mathematics for ACT

1. What is the purpose of logarithms in the ACT exam?
Ans. Logarithms are commonly tested on the ACT to assess a student's understanding of exponential functions and their properties. They are used to solve equations involving exponents and to manipulate exponential expressions.
2. How can logarithms be used to simplify expressions on the ACT exam?
Ans. Logarithms can be used to condense complex exponential expressions into simpler forms by using the properties of logarithms, such as the power rule and product rule. This simplification process can help in solving equations more efficiently.
3. Are there specific strategies for tackling logarithm questions on the ACT exam?
Ans. Yes, some strategies for approaching logarithm questions on the ACT include converting between exponential and logarithmic forms, applying logarithmic properties, and using the laws of exponents to simplify expressions. Practice and familiarity with logarithmic functions can also improve performance on these types of questions.
4. What are some common mistakes to avoid when working with logarithms on the ACT exam?
Ans. Common mistakes when dealing with logarithms on the ACT include forgetting to apply the rules of logarithms correctly, misinterpreting the base of the logarithm, and overlooking the possibility of using logarithms to simplify complex expressions. It is important to carefully follow the steps and rules when working with logarithms.
5. How can understanding logarithms help in solving real-world problems on the ACT exam?
Ans. Logarithms are often used in real-world scenarios involving exponential growth or decay, such as population growth, radioactive decay, and financial investments. Understanding logarithms can help students analyze and solve these types of problems effectively on the ACT exam.
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