Transcript 5.3
5 Exponential and Logarithmic Functions Exponential and Logarithmic Functions 5.3 Logarithms Objectives • Switch between exponential and logarithmic form of equations. • Evaluate logarithmic expressions. • Solve logarithmic equations. • Apply the properties of logarithms to simplify expressions. Logarithms Definition 5.2 If r is any positive real number, then the unique exponent t such that bt = r is called the logarithm of r with base b and is denoted by logb r. Logarithms According to Definition 5.2, the logarithm of 16 base 2 is the exponent t such that 2t = 16; thus we can write log2 16 = 4. Likewise, we can write log10 1000 = 3 because 103 = 1000. In general, Definition 5.2 can be remembered in terms of the statement logb r = t is equivalent to bt = r Logarithms Evaluate log10 0.0001. Example 1 Logarithms Example 1 Solution: Let log10 0.0001 = x. Changing to exponential form yields 10x = 0.0001, which can be solved as follows: 10x = 0.0001 10x = 10-4 1 1 0.0001 4 104 10,000 10 x = -4 Thus we have log10 0.0001 = -4. Properties of Logarithms Property 5.3 For b > 0 and b 1, logb b = 1 and logb 1 = 0 Properties of Logarithms Property 5.4 For b > 0, b 1, and r > 0, blogb r = r Properties of Logarithms Property 5.5 For positive numbers b, r, and s, where b 1, logb rs = logb r + logb s Properties of Logarithms Example 5 If log2 5 = 2.3219 and log2 3 = 1.5850, evaluate log215. Properties of Logarithms Example 5 Solution: Because 15 = 5 · 3, we can apply Property 5.5 as follows: log2 15 = log2(5 · 3) = log2 5 + log2 3 = 2.3219 + 1.5850 = 3.9069 Properties of Logarithms Property 5.6 For positive numbers b, r, and s, where b 1, r logb logb r logb s s Properties of Logarithms Property 5.7 If r is a positive real number, b is a positive real number other than 1, and p is any real number, then logb rp = p(logb r)