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(Read as log base a of x) CONVERTING BETWEEN LOGS AND EXPONENTIAL EXPRESSIONS: Convert the following functions into either exponential or logarithmic form: Exponential equationLogarithmic equation35=243log3243=510-1 = 0.1log64(4)=1/345 = 1024log8512=365 = 7776Evaluating Logs with the Definition of Logarithm: Logarithm with base 10 such as  EMBED Equation.3  is called common logarithm and is written as  EMBED Equation.3  (base 10 is invisible) Common Logarithm can be evaluated on the calculator. **NOTE: LOG key on your calculator indicates common log with base 10 only!** Examples: Evaluate the following functions using the definition of logs.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  2f(x) = 32 25 = 32 f(x) = 5 Examples: Evaluate the following expressions, using the definition of logs: 5. log5(25). 6. log2(8). 7. log64(4). 8. log6(6). 9. log4(16). 10. log2(0). 5y=25 52 = 25 y=2 Important Logarithmic Identities are:    EMBED Equation.3  because  EMBED Equation.3   EMBED Equation.3  because  EMBED Equation.3   EMBED Equation.3  because  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  **there is no way we can raise a positive base to some power and end up with zero**  EMBED Equation.3   EMBED Equation.3  ( x = y Evaluating Logs that do not simplify easily There is a formula that you can use with logarithms that do not simplify easily. For instance, try to evaluate EMBED Equation.DSMT4 . Change of Base Formula: For any positive real numbers  EMBED Equation.DSMT4  This formula puts both logs into the base of 10, which your calculator can help you evaluate.  EMBED Equation.DSMT4  = log108 / log103 This can be evaluated using the log button on your calculator (because its base 10!) log38 = 0.90/0.48 = 1.89 Examples: Evaluate the following expressions, using the change of base formula. a. log49 b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  Where do we use logarithmic functions? Richter Scale The magnitude of an earthquake is  INCLUDEPICTURE "http://www.sosmath.com/algebra/logs/log5/log56/img2.gif" \* MERGEFORMATINET  where I is intensity of the earthquake and S is intensity of a standard earthquake Decibels (dB) "Loudness" is measured in decibels. The formula for the loudness of a sound is given by "dB = 10log[ I I0 ]" where I0 is the intensity of "threshold sound", or sound that can barely be perceived. The pH scale - Chemists define the acidity or alkalinity of a substance according to the formula "pH = log[H+]" where [H+] is the hydrogen ion concentration, measured in moles per liter. Mixed Practice: Evaluate the following expressions/equations, using the definitions and properties of logs. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.3  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4 =x 11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  15.  EMBED Equation.3  16.  EMBED Equation.3  17.  EMBED Equation.3  18.  EMBED Equation.3  19.  EMBED Equation.3  20.  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