📐 General Form of Logarithm and Its Definition

Sahabat Smaridasa — Logarithm is the inverse (reverse) of exponentiation (powers). This means that logarithms are still closely related to exponents, especially when we discuss the inverse of a function. In general, the logarithmic form consists of three parts: the base, the numerus, and the result of the logarithm. Logarithms are very important not only in mathematics, but also in other fields such as chemistry (related to oxidation numbers) and other areas involving growth functions.

General Form of Logarithm and Its Definition is the basic material that we must master, especially in the introduction stage to logarithms. By knowing the general form and definition of logarithms, it will make it easier for us to learn subsequent material related to logarithms. In fact, there is a UI (University of Indonesia) entrance exam question where the solution only uses the definition of logarithm, but the solution is not as easy as we imagine because it requires further analysis.

📖 General Form and Definition

$ {}^a \log b = c \iff b = a^c $

or

$ {}^a \log b = c \iff a^c = b $

with $ a, b, c $ real numbers ($ \mathbb{R} $) and $ a > 0, a \neq 1, b > 0 $

Description:
$ a $ is called the base
$ b $ is called the numerus
$ c $ is called the result of the logarithm

Here are examples of logarithms to help you understand the material better.

Example

Determine the result of the following logarithmic forms:
(i) $ {}^2 \log 4 $    (ii) $ {}^3 \log 81 $    (iii) $ {}^5 \log 125 $    (iv) $ \log 1000 $

Solution:

Based on the general form of logarithm and its definition:

(i) $ {}^2 \log 4 = 2 $, because $ 2^2 = 4 $

(ii) $ {}^3 \log 81 = 4 $, because $ 3^4 = 81 $

(iii) $ {}^5 \log 125 = 3 $, because $ 5^3 = 125 $

(iv) $ \log 1000 = {}^{10} \log 1000 = 3 $, because $ 10^3 = 1000 $

Note: For logarithms with base 10, the number 10 does not need to be written. For example, $ {}^{10} \log a $ can be written simply as $ \log a $ and its value remains the same. $ \log a $ means it has base 10.

To be able to solve logarithm problems, the definition alone is not enough. We must also master the properties of logarithms well, because usually every logarithm problem uses the properties of logarithms. In addition to using logarithm properties, determining logarithm values can also be done using a calculator and mathematical tables.

The materials that we will discuss in logarithmic form are properties of logarithms, logarithmic functions, logarithmic equations, and logarithmic inequalities.

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💡 Tip: The best way to master logarithms is to first understand the relationship between logarithms and exponents. Practice converting between exponential and logarithmic forms until it becomes natural.