1. Solving Logarithmic Equations I.. Relationship between Exponential and Logarithmic Equations. A) Logs and Exponentials are INVERSES of each other. 1) That means they cancel each other out. B) To solve equations that have a variable in the exponent, you convert them into logarithmic form. C) To solve equations that have a logarithm in them, you convert them into exponential form. D) Remember that ln is log e. You might have to rewrite it in its other form to solve for x.

2. II. Solving Logarithmic Equations Using Exponential Inverse. *A) Condense the logs (if needed). Can only have ONE log. B) Circle the log base # part. C) Isolate the circled part. 1) Steps for Solving Equations notes. 2) Now you have log base m = n. D) Convert into exponential form. ( remember ln = log e ) 1) base n = m ( ln m = n is log e m = n  e n = m ) E) Evaluate using a calculator. F) Solve for the variable (if needed). Solving Logarithmic Equations

3. Examples: Solve for the variable. 1) 21 + 9 log 7 x = 3 21 + 9 log 7 x = 3 (circle the log part) 9 log 7 x = –18 (isolate the circle) (subtract 21) log 7 x = –2 ( ÷ by 9 ) 7 –2 = x (pop into exponent form) 1 / 49 = x (evaluate) Solving Logarithmic Equations

4. 2) 2 ln 3x – 5 = 15 2 ln 3x – 5 = 15 (circle the log part) 2 ln 3x = 20 (isolate the circled part) ( add 5 ) ln 3x = 10 ( ÷ by 2 ) log e 3x = 10 ( ln is log e ) e 10 = 3x (pop into exponential form) 22026.466 = 3x (solve for x) 7342.155 = x ( ÷ by 3 ) Solving Logarithmic Equations