7 4 Properties of Logarithms Warm Up Simplify

7 -4 Properties of Logarithms Warm Up Simplify. 1. (3– 2)(35) 3. (73)5 2. Write in exponential form. 4. logx x = 1 5. 0 = logx 1 LEARNING GOALS – LESSON 7. 4. 1: Use properties to simplify logarithmic expressions. 7. 4. 2: Translate between logarithms in any base. Remember that to multiply powers with the same base, you add exponents. Log Rule: Product Property logbmn = logbm + logbn Example: log 32 x = log 32 + log 3 x Example 1: Adding Logarithms A. Express log 64 + log 69 as a single logarithm. Simplify.

7 -4 Properties of Logarithms Express as a single logarithm. Simplify, if possible. B. log 5625 + log 525 Caution Just as a 5 b 3 cannot be simplified, logarithms must have the same base to be simplified. Remember that to divide powers with the same base, you subtract exponents Log Rule: Quotient Property Example: Example 2: Subtracting Logarithms Express as a single logarithm. Simplify, if possible. A. log 5100 – log 4 5 B. log 749 – log 7 7

7 -4 Properties of Logarithms Log Rule: Power Property Example: Example 3: Simplifying Logarithms with Exponents Express as a product. Simplify, if possible. A. log 2326 C. log 104 B. log 8420 D. log 5252 E. log 2 ( ½ )5 Exponential and logarithmic operations undo each other since they are ________ operations. INVERSE PROPERTIES Log Rule: Inverse Properties base b such that b > 0 and b≠ 1 Example:

7 -4 Properties of Logarithms Example 4: Recognizing Inverses Simplify each expression. A. log 3311 B. log 381 D. log 100. 9 C. 5 log 510 E. 2 log 2(8 x) NOTE: YOUR CALCULATOR ONLY CAN COMPUTE LOGS IN BASE _____ and_____. You can change a logarithm in one base to a logarithm in another base with the following formula . Log Rule: Change of Base Example: *You may choose the a value Example 5: Changing the Base of a Logarithm Evaluate log 328. Method 1 Change to base 10 Method 2 Change to base 2, because both 32 and 8 are powers of 2.

7 -4 Properties of Logarithms B. Evaluate log 927. Method 1 Change to base 10. Method 2 Change to base 3, because both 27 and 9 are powers of 3. C. Evaluate log 816. Method 2 Change to base _____, because both 27 and 9 are powers of _____. Logarithmic scales are useful for measuring quantities that have a very wide range of values, such as the intensity (loudness) of a sound or the energy released by an earthquake. This is applicable because they go by powers of 10. Example 6: Geology Application The tsunami in December 2004 started by an earthquake with magnitude 9. 3 How many times as much energy did this earthquake release compared to the 6. 9 -magnitude earthquake that struck San Francisco in 1989? Step 1: Plug in 9. 3 and 6. 9 for M. Step 2: Solve for E. Step 3: Compare your answers.