Properties of Logarithms Section 11.6 Properties of Logarithms

Objectives Use the four basic properties of logarithms. Use the product rule for logarithms. Use the quotient rule for logarithms. Use the power rule for logarithms. Write logarithmic expressions as a single logarithm. Use the change-of-base formula. Use properties of logarithms to solve application problems.

Objective 1: Use the Four Basic Properties of Logarithms. Properties of Logarithms: For all positive numbers b, where b ≠ 1, We can use the definition of logarithm to prove that these properties are true. Properties 3 and 4 also indicate that the composition of the exponential and logarithmic functions (in both directions) is the identity function. This is expected, because the exponential and logarithmic functions are inverse functions.

EXAMPLE 1 Simplify each expression: Strategy We will compare each logarithmic expression to the left side of the previous four properties of logarithms. Why When we get a match, the property will provide the answer.

EXAMPLE 1 Simplify each expression: Solution

Objective 2: Use the Product Rule for Logarithms. The logarithm of a product equals the sum of the logarithms of the factors. For all positive real numbers M, N, and b, where b ≠ 1, logb MN = logb M + logb N Read as “the log base b of M times N equals the log base b of M plus the log base b of N.”

EXAMPLE 2 Write each expression as a sum of logarithms. Then simplify, if possible. Strategy In each case, we will use the product rule for logarithms. Why We will use the product rule as each logarithmic expressions has the form logbMN.

EXAMPLE 2 Write each expression as a sum of logarithms. Then simplify, if possible. Solution

Objective 3: Use the Quotient Rule for Logarithms. The Quotient Rule for Logarithms The logarithm of a quotient equals the difference of the logarithms of the numerator and denominator. For all positive real numbers M, N and b where b ≠ 1, Read as “the log base b of M divided by N equals the log base b of M minus the log base b of N.”

EXAMPLE 3 Write each expression as a difference of logarithms. Then simplify, if possible. Strategy In both cases, we will apply the quotient rule for logarithms. Why We use the quotient rule because each of the logarithmic expressions has the form

EXAMPLE 3 Write each expression as a difference of logarithms. Then simplify, if possible. Solution

Objective 4: Use the Power Rule for Logarithms. The Power Rule for Logarithms The logarithm of a number raised to a power equals the power times the logarithm of the number. For all positive real numbers M, and b where b ≠ 1, and any real number p, logb M p = p logb M Read as “the log base b of M to the p power equals p times the log base b of M.”

EXAMPLE 5 Write each logarithm without an exponent or a square root: Strategy In each case, we will use the power rule for logarithms. Why We use the power rule because log5 62 has the form logb M p, as will if we write

EXAMPLE 5 Write each logarithm without an exponent or a square root: Solution

Objective 5: Write Logarithmic Expressions as a Single Logarithm. Properties of Logarithms Summarized

EXAMPLE 7 Write each logarithmic expression as one logarithm: Strategy In part (a), we will use the power rule and product rule for logarithms in reverse. In part (b), we will use the power rule, the quotient rule, and the product rule for logarithms in reverse. Why We use the power rule because we see expressions of the form p logb M. The + symbol between logarithmic terms suggests that we use the product rule and the − symbol between such terms suggests that we use the quotient rule.

EXAMPLE 7 Solution Write each logarithmic expression as one logarithm: a. We begin by using the power rule on both terms of the expression.

EXAMPLE 7 Solution Write each logarithmic expression as one logarithm: b. The first and third terms of this expression can be rewritten using the power rule of logarithms. Note that the base of each logarithm is b. We do not need to know the value of b to apply properties of logarithms.

EXAMPLE 7 Write each logarithmic expression as one logarithm: Solution

Objective 6: Use the Change-of-Base Formula. Change-of-Base Formula: For any logarithmic bases a and b, and any positive real number x, This formula converts a logarithm of one base to a ratio of logarithms of a different base. Most calculators can find common logarithms (base 10) and natural logarithms (base e). If we need to find a logarithm with some other base, we can use the following change-of-base formula.

EXAMPLE 9 Find log3 5 Strategy To evaluate this base-3 logarithm, we will substitute into the change-of-base formula. Why We assume that the reader does not have a calculator that evaluates base-3 logarithms (at least not directly). Thus, the only alternative is to change the base.

EXAMPLE 9 Solution Find log3 5 To find log3 5, we substitute 3 for b, 10 for a, and 5 for x in the change-of-base formula and simplify: To four decimal places, log3 5 = 1.4650.

EXAMPLE 9 Solution Find log3 5 We can also use the natural logarithm function (base e) in the change-of-base formula to find a base-3 logarithm. We obtain the same result.

Objective 7: Use Properties of Logarithms to Solve Application Problems. In chemistry, common logarithms are used to express how basic or acidic a solution is. The more acidic a solution, the greater the concentration of hydrogen ions. (A hydrogen ion is a positively charged hydrogen atom missing its electron.) The concentration of hydrogen ions in a solution is commonly measured using the pH scale. The pH of a solution is defined as follows. If [H+] is the hydrogen ion concentration in gram-ions per liter, then pH = –log [H+]

EXAMPLE 10 pH Meters. One of the most accurate ways to measure pH is with a probe and meter. What reading should the meter give for lemon juice if it has a hydrogen ion concentration [H+] of approximately 6.2  10−3 gram-ions per liter? Strategy We will substitute into the formula for pH and use the power rule for logarithms to simplify the right side. Why After substituting 6.2  10−3 for [H+] in –log [H+], the resulting expression will have the form logb M p.

EXAMPLE 10 pH Meters. One of the most accurate ways to measure pH is with a probe and meter. What reading should the meter give for lemon juice if it has a hydrogen ion concentration [H+] of approximately 6.2  10−3 gram-ions per liter? Solution Since lemon juice has approximately 6.2  10−3 gram-ions per liter, its pH is

EXAMPLE 10 pH Meters. One of the most accurate ways to measure pH is with a probe and meter. What reading should the meter give for lemon juice if it has a hydrogen ion concentration [H+] of approximately 6.2  10−3 gram-ions per liter? Solution The meter should give a reading of approximately 2.2.