LEARNING GOALS – LESSON 7.4 Example 1: Adding Logarithms Warm Up Simplify. 2. 1. (3–2)(35) 3. (73)5 Write in exponential form. 4. logx x = 1 5. 0 = logx1 LEARNING GOALS – LESSON 7.4 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 Example: logbmn = logbm + logbn log32x = log32 + log3x Example 1: Adding Logarithms A. Express log64 + log69 as a single logarithm. Simplify.

Example 2: Subtracting Logarithms Express as a single logarithm. Simplify, if possible. B. log5625 + log525 Just as a5b3 cannot be simplified, logarithms must have the same base to be simplified. Caution 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. log5100 – log54 B. log749 – log77

Example 3: Simplifying Logarithms with Exponents Log Rule: Power Property Example: Example 3: Simplifying Logarithms with Exponents Express as a product. Simplify, if possible. A. log2326 B. log8420 C. log104 D. log5252 E. log2 ( 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:

Log Rule: Change of Base Example: Example 4: Recognizing Inverses Simplify each expression. A. log3311 B. log381 C. 5log510 D. log100.9 E. 2log2(8x) 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 log328. Method 2 Change to base 2, because both 32 and 8 are powers of 2. Method 1 Change to base 10

Example 6: Geology Application B. Evaluate log927. Method 1 Change to base 10. Method 2 Change to base 3, because both 27 and 9 are powers of 3. C. Evaluate log816. 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 in1989? Step 1: Plug in 9.3 and 6.9 for M. Step 2: Solve for E. Step 3: Compare your answers.