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Math 152B - Cass Gustafson – 9.5 Exponential and Logarithmic Functions Properties of Logarithms I. Basic Logarithmic Properties Based on the Definition of a Logarithm A. If b > 0 and b ≠ 1, then 1) logb b = 1 because … 2) logb b x = x because … 3) logb 1 = 0 because … B. If b = 10, then 1) log 1 = 0 because … 2) log 10 = 1 because … C. If b = e, then 1) ln 1 = 0 because … 3) ln e = 1 because … 4) blogb x = x because … x 3) log 10 = x because … 4) 10 log x = x because … x 2) ln e = x because … 4) e ln x = x because … Ex. 1: Evaluate the following. Hint: set an expression equal to x if you’re not sure what rule applies. 3 a) log9 9 b) log 10 c) log8 1 d) ln e e) 3log3 17 f) g) log7 78 h) 10 ln 1 log 7 II. Properties of Logarithms Based on the Properties of Exponents Since a logarithm is an exponent, the properties of exponents are also properties of logarithms. A. The Product Rule: The logarithm of a product is the sum of the logarithms. logb ( MN ) = logb M + logb N for positive real numbers M, N, and b, b ≠ 1. Ex. 2: Expand the following expressions using the Product Rule. a) log6 ( 7 ⋅11) b) log (100x ) B. The Quotient Rule: The logarithm of a quotient is the difference of the logarithms. M logb = logb M − logb N N for positive real numbers M, N, and b, b ≠ 1. Ex. 3: Expand the following expressions using the Quotient Rule. 23 a) log8 x e5 b) ln 11 C. The Power Rule: The logarithm of a number with an exponent is the product of the exponent and the logarithm of that number. logb Mp = p ⋅ logb M for positive real numbers M and b, b ≠ 1, and any real number p Ex. 4: Expand the following expressions using the Power Rule. a) log6 89 Gustafson – 9.5 b) ln 3 x 2 III. Expanding Logarithmic Expressions: When we use the above rules to write a single logarithm as a sum, difference, or product of other numbers and/or logarithms, we say that we are expanding a logarithmic expression. Ex. 5: Use the logarithmic properties to expand each expression and write it in terms of the logarithms of x, y, and z. x a) logb yz x3 y2 b) logb 4 4 z IV. Condensing Logarithmic Expressions: When we use the rules of logarithms to write the sum or difference of two or more logarithmic expressions as a single logarithmic expression, we say that we are condensing a logarithmic expression. Ex. 6: Write as a logarithm of a single quantity. a) −2 logb x − 3 logb y + logb z b) 2 logb x + 1 log y − 2 logb ( x − y ) 2 b Ex. 7: Given that log 5 ≈ 0.6990 and log 6 ≈ 0.7782, find approximations for the following without using a calculator. a) log 30 b) log 25 c) log 180 1 d) log 2 Gustafson – 9.5 3 V. The Change-of-Base Property: The logarithm of M with base b is equal to the logarithm of M with any new base divided by the logarithm of b with that new base. logb M = loga M for positive real numbers M, a and b, a ≠ 1, b ≠ 1. loga b logb M = logM lnM = for positive real numbers M and b, b ≠ 1. logb lnb Ex. 8: Find each logarithm to four decimal places. a) log7 2506 b) logπ e VI. Properties of Equality for Exponentials and Logarithms These properties will be used when solving exponential and logarithmic equation. M A. b = b B. M = N N M = N (for b > 0 and b ≠ 1) logb M = logb N (for all positive M and N) VII. Applications A. pH of a Solution: The pH of a solution is a measure of its acidity, that is, a measure of the concentration of hydrogen ions in the solution. The more acidic a solution, the greater the concentration of hydrogen ions. The concentration is indicated by the pH scale, which is defined by the following equation… pH = –log [H+] , where [H+] is the hydrogen ion concentration in gram-ions per liter. Ex. 9: Find the hydrogen ion concentration of a saturated solution of calcium hydroxide whose pH is 13.2. B. In physiology, the relationship between the loudness and the intensity of sound is based on the Weber-Fechner law, which is defined by the following equation… L = k ln I , where L is the apparent loudness of a sound, I is the actual intensity, and k is a constant. Ex. 10: If the intensity of a sound is tripled, find the apparent change in loudness. Gustafson – 9.5 4