7.5: Apply Properties of Logarithms • • Objectives Use properties to simplify logarithmic expressions. Translate between logarithms in any base. Common Core Standards: • A-REI-2, F-BF-5 • • Assessments: Define all vocab for this section Do worksheet 7-5 Remember that to multiply powers with the same base, you add exponents. Expand the Logarithm • Log7 6x Express log64 + log69 as a single logarithm. Then Solve. log64 + log69 log6 (4 9) log6 36 2 To add the logarithms, multiply the numbers. Simplify. Think: 6? = 36. Express as a single logarithm. Solve, if possible. log5625 + log525 Given: Log 3= .477 What is log 21? What is log 63? Log 7= .845 The property above can also be used in reverse. Caution Just as a5b3 cannot be simplified, logarithms must have the same base to be simplified. Remember that to divide powers with the same base, you subtract exponents Because logarithms are exponents, subtracting logarithms with the same base is the same as finding the logarithms of the quotient with that base. Express log5100 – log54 as a single logarithm. Simplify, if possible. log5100 – log54 Express log749 – log77 as a single logarithm. Simplify, if possible. Given: Log 3= .477 What is log 7? Log 21= 1.322 Because you can multiply logarithms, you can also take powers of logarithms. Power property of logarithms Simplify • Log3 1000 • Log2 x3 Express as a product. Simplify, if possible. A. log2326 6log232 B. log8420 20log84 EXAMPLE 2 Expand a logarithmic expression Expand log 6 5x3 y SOLUTION log 6 5x3 = log 6 5x3 – log 6 y y = log 6 5 + log 6 x3 – log 6 y = log 6 5 + 3 log 6 x – log 6 y Quotient property Product property Power property Exponential and logarithmic operations undo each other since they are inverse operations. Simplify each expression. a. log3311 b. log381 c. 5log510 a. Simplify log 100.9 b. Simplify 2log2(8x) Most calculators calculate logarithms only in base 10 or base e (see Lesson 7-4). You can change a logarithm in one base to a logarithm in another base with the following formula. • Change of Base formula Loga b = • Log3 8 = log 8 log 3 log 𝑏 log 𝑎 =1.893 or or loga b = ln 𝑎 ln 𝑏 log3 8 = ln 8 ln 3 =1.893 Evaluate log328. Evaluate log927.