Q1-GenMath_Module-7-Wk7.pdf

GENERAL MATHEMATICS Module 7: Logarithmic Functions, Equations, and Inequalities Presented by Ms. Jiezyl Jamaica M. Aquino 1. represent real-life situations using logarithmic functions (M11GM-Ih-1), 2. distinguish logarithmic function, logarithmic equation, and logarithmic...

GENERAL MATHEMATICS Module 7: Logarithmic Functions, Equations, and Inequalities Presented by Ms. Jiezyl Jamaica M. Aquino 1. represent real-life situations using logarithmic functions (M11GM-Ih-1), 2. distinguish logarithmic function, logarithmic equation, and logarithmic MELCs inequality (M11GM-Ih-2); and 3.solve logarithmic equations and inequalities (M11GM-Ih- i-1). Representation of Logarithmic Function to Real-Life Situation  LOGARITHMIC FUNCTION A logarithm is defined as the exponent that indicates the power to which a base number is raised to produce a given number. The logarithm of a with base b is denoted by 𝒍𝒐𝒈𝒃𝒂 and is defined as 𝒄 = 𝒍𝒐𝒈𝒃𝒂 if and only if 𝒂 = 𝒃. 𝒄  EXAMPLE 𝒂= 𝒃𝒄 𝒄 = 𝒍𝒐𝒈𝒃𝒂 𝟖= 𝟐𝟑 𝟑 = 𝒍𝒐𝒈𝟐𝟖 Answer 𝒍𝒐𝒈𝟐𝟖 = 𝟑 Exponent Base 𝑬𝒙𝒑𝒐𝒏𝒆𝒏𝒕𝒊𝒂𝒍 𝑭𝒐𝒓𝒎 𝑳𝒐𝒈𝒂𝒓𝒊𝒕𝒉𝒎𝒊𝒄 𝑭𝒐𝒓𝒎 EXPONENT 𝒃𝒄 =𝒂 𝒄 = 𝒍𝒐𝒈𝒃𝒂 BASE Note that in both the logarithmic and exponential forms, b is the base. In the exponential form, c is an exponent; this implies that the logarithm is actually an exponent. Hence, logarithmic and exponential functions are inverses. In dealing with logarithms, it is important to note the following.  LOGARITHMIC FUNCTION A logarithmic function expresses a relationship between two variables (such as x and y) and can be represented by a table of values or a graph. The logarithmic function is the function 𝒚 = 𝒍𝒐𝒈𝒃𝒙 where b is any number such that 𝒃 > 𝟎, 𝒃 ≠ 𝟏, and 𝒙 > 𝟎. LOGARITHMIC FUNCTION LOGARITHMIC FUNCTION, EQUATION, & INEQUALITY  Rewriting Logarithmic Equation to its Exponential Form  Rewriting Logarithmic Equation to its Exponential Form 𝒍𝒐𝒈𝟑𝟖𝟏 𝒃𝒄 =𝒂 𝟑? = 𝟖𝟏 𝟑𝟒 = 𝟖𝟏 or 𝒍𝒐𝒈𝟑𝟖𝟏 = 𝟒 Rewriting Logarithmic Equation to its Exponential Form & vice versa  PROPERTIES OF LOGARITHMS Recall that 𝒚 = 𝒍𝒐𝒈𝒃𝒙 is equivalent to 𝒚 𝒃 = 𝒙 for 𝒙 > 𝟎, 𝒃 ≠ 𝟏, and 𝒙 > 𝟎.  PROPERTIES OF LOGARITHMS Property 1. 𝒍𝒐𝒈𝒃𝟏 = 𝟎. EXAMPLE:  PROPERTIES OF LOGARITHMS Property 2. 𝒍𝒐𝒈𝒃𝒃 = 𝟏. EXAMPLE:  PROPERTIES OF LOGARITHMS Property 3. Product Property of Logarithms PROPERTIES OF LOGARITHMS Property 3. Product Property of Logarithms EXAMPLE:  PROPERTIES OF LOGARITHMS Property 4. Quotient Property of Logarithms PROPERTIES OF LOGARITHMS Property 4. Quotient Property of Logarithms EXAMPLE: PROPERTIES OF LOGARITHMS Property 5. Power Property of Logarithms PROPERTIES OF LOGARITHMS Property 5. Power Property of Logarithms EXAMPLE: PROPERTIES OF LOGARITHMS Property 6. Let b and M be positive real numbers with b ≠ 1. PROPERTIES OF LOGARITHMS Property 6. Let b and M be positive real numbers with b ≠ 1. EXAMPLE: PROPERTIES OF LOGARITHMS Property 6. Let b and M be positive real numbers with b ≠ 1. EXAMPLE: Use the change-of-base formula to find an approximation up to four decimal places for each of the logarithm expressions.  Solving Logarithmic Equations  Solving Logarithmic Inequalities THANK YOU!