SAT Logarithms: Definition, Types, Properties, Formulas, Examples & Applications - Testbook

Logarithms might sound tricky at first, but they’re actually a super cool shortcut for working with exponents! Think of them as the flip side of powers—like the undo button for exponentiation. For example, if you know that 10² = 100, then the logarithm flips it around: log₁₀(100) = 2. Pretty neat, right?

Here’s the general rule:
log₍b₎(x) = n means bⁿ = x, where b is the base.

If you're gearing up for exams like the SAT, ACT, PSAT/NMSQT, GED, GRE, GMAT, AP Exams, PERT, Accuplacer, or even the MCAT, knowing how logarithms work will save you tons of time on those algebra and problem-solving questions. In this article, we'll break down exactly what logarithms are, explain the difference between common logs and natural logs, and walk you through the key properties—with plenty of examples—so you can crush those test questions with confidence!

Logarithms

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Historical Background

The concept of Logarithms was first introduced by John Napier in the 17th century. Since then, it has been widely used by scientists, navigators, engineers, and others to simplify complex calculations. In essence, Logarithms are the inverse process of exponentiation.

Defining Logarithms

A logarithm is essentially the power to which a number must be raised to obtain another value. It is an efficient way to express large numbers . A logarithm possesses several essential properties that demonstrate that the multiplication and division of logarithms can also be represented in the form of the addition and subtraction of logarithms.

“The logarithm of a positive real number a with respect to base b, a positive real number not equal to 1 ^{[nb 1]} , is the exponent by which b must be raised to yield a”.

In other words, b ^y = a ⇔ log _b a =y

Where,

“a” and “b” are two positive real numbers
y is a real number
“a” is referred to as the argument, which is inside the log
“b” is known as the base, which is at the bottom of the log.

In simple terms, the logarithm answers the question “How many times a number is multiplied to get the other number?”.

For instance, how many 3’s are multiplied to get the answer 27?

If we multiply 3 three times, we get the answer 27.

Therefore, the logarithm is 3.

The logarithm form is written as follows:

Log ₃ (27) = 3 ….(1)

Therefore, the base 3 logarithm of 27 is 3.

The above logarithm form can also be written as:

3x3x3 = 27

3 ³ = 27 …..(2)

Thus, the equations (1) and (2) both represent the same meaning.

Here are some examples of conversion from exponential forms to logarithms.

Exponents	Logarithms
7 ² = 49	Log ₇ 49 = 2
8 ² = 64	Log ₈ 64 = 2
4 ³ = 64	Log ₄ 64 = 3

Types of Logarithms

Typically, we encounter two distinct types of logarithms, namely:

Common Logarithm
Natural Logarithm

Common Logarithm

The common logarithm, also referred to as the base 10 logarithm, is denoted as log 10 or simply log. For instance, the common logarithm of 1000 is written as log (1000). The common logarithm determines how many times we need to multiply the number 10 to achieve the desired result.

For instance, log (100) = 2

If we multiply the number 10 twice, we obtain the result 100.

Natural Logarithm

The natural logarithm, also known as the base e logarithm, is represented as ln or log e . Here, “e” represents Euler’s constant, approximately equal to 2.71828. For instance, the natural logarithm of 78 is written as ln 78. The natural logarithm determines how many times we need to multiply “e” to achieve the desired result.

For instance, ln (78) = 4.357.

Thus, the base e logarithm of 78 is equal to 4.357.

Logarithmic Properties

Logarithms operate on certain rules. These rules include:

Product rule
Division rule
Power rule/Exponential Rule
Change of base rule
Base switch rule
Derivative of log
Integral of log

Let's take a closer look at each of these properties.

Product Rule

According to this rule, the multiplication of two logarithmic values is equal to the sum of their individual logarithms.

Log _b (mn)= log _b m + log _b n

For instance: log ₃ ( 2y ) = log ₃ (2) + log ₃ (y)

Division Rule

According to this rule, the division of two logarithmic values is equal to the difference between each logarithm.

Log _b (m/n)= log _b m – log _b n

For instance, log ₃ ( 2/ y ) = log ₃ (2) -log ₃ (y)

Exponential Rule

In this rule, the logarithm of m with a rational exponent is equal to the exponent times its logarithm.

Log _b (m ⁿ ) = n log _b m

Example: log _b (2 ³ ) = 3 log _b 2

Change of Base Rule

Log _b m = log _a m/ log _a b

Example: log _b 2 = log _a 2/log _a b

Base Switch Rule

log _b (a) = 1 / log _a (b)

Example: log _b 8 = 1/log ₈ b

Derivative of log

If f (x) = log _b (x), then the derivative of f(x) is given by;

f'(x) = 1/(x ln(b))

Example: Given, f (x) = log ₁₀ (x)

Then, f'(x) = 1/(x ln(10))

Integral of Log

∫log _b (x)dx = x( log _b (x) – 1/ln(b) ) + C

Example: ∫ log ₁₀ (x) dx = x ∙ ( log ₁₀ (x) – 1 / ln(10) ) + C

Other Properties

Here are some other properties of logarithmic functions:

Log _b b = 1
Log _b 1 = 0
Log _b 0 = undefined

Logarithmic Formulas

log _b (mn) = log _b m + log _b n

log _b (m/n) = log _b m – log _b n

Log _b (x y ) = y log _b (x)

Log _b m √n = log _b n/m

m log _b (x) + n log _b (y) = log _b (x ^m y ⁿ )

log _b (m+n) = log _b m + log _b (1+nm)

log _b (m – n) = log _b m + log _b (1-n/m)

Illustrative Examples

Example 1:

Solve log ₃ (81) =?

Solution:

since 3 ⁴ = 3 × 3 × 3 × 3 = 81, 4 is the exponent value and log ₃ (81)= 4.

Example 2:

What is the value of log ₁₀ (1000)?

Solution:

In this case, 10 ³ yields you 1000. So, 3 is the exponent value, and the value of log ₁₀ (1000)= 3

Example 3:

Using the property of logarithms, solve for the value of x for log ₃ x= log ₃ 4+ log ₃ 7

Solution:

By the addition rule, log ₃ 4+ log ₃ 7= log ₃ (4 * 7 )

Log ₃ ( 28 ). Thus, x= 28.

Example 4:

Solve for x in log ₂ x = 5

Solution:

This logarithmic function can be written In the exponential form as 2 ⁵ = x

Therefore, 2 ⁵ = 2 × 2 × 2 × 2 × 2 = 32, x= 32.

Example 5:

Find the value of log ₅ (1/125).

Solution:

Given: log ₅ (1/125)

By using the property,

Log _b (m/n)= log _b m – log _b n

log ₅ (1/125) = log ₅ 1 – log ₅ 125

log ₅ (1/125) = 0 – log ₅ 5 ³

log ₅ (1/125) = -3log ₅ 5

log ₅ (1/125) = -3 (1) [By using the property log _a a = 1)

log ₅ (1/125) = -3.

Hence, the value of log ₅ (1/125) = -3

Practical Applications

Logarithms are extensively used in various fields of science and technology. Logarithmic calculators have made our calculations much easier. They are also used in surveying and celestial navigation. Additionally, they are used to measure the loudness (decibels), the intensity of earthquakes in terms of the Richter scale, in radioactive decay, and to measure acidity (pH= -log10[H+]), among other applications.

Conclusion

To wrap it up, logarithms aren’t just abstract math concepts—they’re powerful tools that pop up everywhere, from solving tricky test questions to real-life science and tech applications. Whether you're prepping for the SAT, ACT, GED, or even the MCAT, mastering logarithms will give you a serious edge. Once you get the hang of their rules and properties, you'll see how they simplify complex problems and make calculations a breeze!

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