Exponential Equations, Inequalities, and Functions, Study notes of Mathematics

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Lesson 6 : Logarithmic Equations and Inequalities

Lesson 6: Logarithmic Equations and Inequalities Introduction Logarithmic equations are those containing the logarithmic logarithmic expression log, x, where 2 > 0. Like exponential equations, these equations are jf situations such as on the values of increase or decrease in population, interests, and pH level. Since there are measurements such used to describe real as that af growth and decay, the study of exponential and logarithmic equations is very important. Definition of Logarithms Let a and b be positive real numbers such that b # 1. The logarithm of a with base b, denoted by log, a is defined as the number such that b"* = a. That is, log, a is the exponent that b must be raised to produce a. Logarithms and exponents allow us to express the same relationship in two different ways. Example 1: Logarithmic Form Exponential Form Jog, 32-5 —32 Jog, 5 =1 s=5 Jog: 16 = — Gy“ =16 Jog, 1 — P-1 Exponential and Logarithmic Forms Logarithmic form: log, a = Exponential form: b° = In both the logarithmic and exponential forms, b is the base. In the exponential form, c is an exponent. But ¢ = log, a .This implies that the logarithm is actually an exponent. In the logarithmic form, ais called the argument, b is the base, and c is the exponent. In the logarithmic form logy, a, the argument a cannot be negative. For example, log, (—8) is not defined since 2 raised to any exponent will never result to a negative number. The value of log, « can be negative. For example, log, (33) — —8 because 5 — 55. ‘Common logarithms are logarithms with bese 10; the base 10 is usually omitted when writing common logarithms. This means that log is a short notation for login &. ‘As mentioned in the previous lessons, the number e (which is approximately 2.71828) has important applications in mathematics. Logarithms with base e are called natural logarithms, and are denoted by “In”. In other words, In z is another way of writing log, 2. Example 2: Rewrite the following exoonential equations in logarithmic form, whenever possible. Lesson 6: Logarithmic Equations and Inequalities