Logarithm Rules and Properties

The common laws of logarithms, generally known as the “log rules,” will be covered in this lecture.

These seven (7) log principles come in handy for expanding, condensing, and solving logarithmic problems. 

I would also urge that you review and grasp the exponent principles, as the inverse of a logarithmic function is an exponential function. They always go together, believe me.

Log’s Characteristics

The features of log are used to compress numerous logarithms into a single logarithm or to expand a single logarithm into multiple logarithms. Exponents are written in a logarithmic format. As a result, logarithm attributes are derived from exponent properties.

Let us understand the properties of log, as well as their proofs, and use these qualities to solve a few problems.

What are Log Properties?

The rules of logarithms, which are derived from the exponent rules, are the properties of log. These logarithmic features are used to simplify logarithmic statements and solve logarithmic problems. The following are four significant logarithmic properties:

loga mn is equal to loga m Plus loga n. (product property)

loga m/n = loga m – loga n (quotient property)

logₐ mn = n logₐ m (power property)

(log a) / (log b) = logb a (change of base property)

Apart from these, there are several other aspects of logarithms that can be deduced directly from the exponent rules and the logarithm definition (which is ax = m ⇔ logₐ m = x).

a0 = 1 ⇒ logₐ 1 = 0

a1 = a ⇒ logₐ a = 1

alogₐ x = x

Properties of Natural Logs

The natural logarithm is simply a logarithm with base “e.” i.e., logₑ = ln. All of the above properties are expressed in terms of “log” and are applicable to any base; thus, all of the above properties apply to natural logs as well. The natural logarithmic qualities are listed below.

• ln 1 = 0
• ln e = 1
• ln (mn) = ln m + ln n
• ln (m/n) = ln m – ln n
• ln mn = n ln m
• eln x = x

Log Product Property

The sum of logs is used to express the logarithm of a product using the product property of logarithms. Let’s figure out what the product property is: loga mn = loga m + loga n.

Log’s Quotient Property

The difference of logs is used to express the logarithm of a quotient using the quotient property of logarithms. Let’s look at how to get the quotient property: loga m/n = loga m – loga n.

Logarithms’ Power Property

The logarithm’s power property states that loga mn = n loga m. It signifies that the argument’s exponent can be pushed in front of the log.

Changes to Log’s Base Property

The base property change is logb a = (log a) / (log b). It indicates that logb a is the quotient of two logarithms (log a)/(log b), where both logs must have the same base (say c). We know that the calculator has two buttons for evaluating logarithms. The first is log (base 10) and the second is ln (base ‘e’). What if we need to calculate a logarithm with a different base, such as log5 2? This characteristic comes in handy when calculating logarithms. When the base property is changed to log5 2, we obtain

log₅ 2 = (log 2) / (log 5)

= (0.3010) / (0.6990) ≈ 0.4306

Important Logarithmic Properties Notes

The logarithmic properties apply to any log, regardless of base. i.e., they can be used with log, ln, (or) log a.

Logarithms have three important properties:

log (m/n) = log m – log n log mn = log m + log n log (m/n) = log m – log n

Regardless of the basis, log mn = n log m log 1 = 0.

To extend or compress logarithms, logarithmic characteristics are used.

Conclusion

The rules of logarithms, which are derived from the exponent rules, are the properties of log. These logarithmic features are used to simplify logarithmic statements and solve logarithmic problems.

The majority of scientific calculators can only calculate logarithms in base 10 (log(x) for common logarithms and base e (ln(x) for natural logarithms) (the reason why the letters l and n are backwards is lost to history). The number e is an irrational number (like pi) with a non-repeating string of decimals going to infinity.