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How Do You Evaluate a Logarithm?

Evaluate log3 81.

Summary

  1. 'log' stands for 'logarithm'
  2. 'b' stands for 'base'
  3. 'x' represents the exponent
  4. y = bx is in exponential form
  5. x = logby is in logarithmic form
  6. y = bx and x = logby are equivalent equations, so you can convert between them
  7. The base of the logarithm is the same as the base of the exponent
  8. The base is the small number down and to the right of 'log'
  9. The number the logarithm is equal to is the exponent in the exponential expression
  10. 3x = 81 is asking 'To what power do we raise 3 to get 81?'

Notes

    1. A logarithm is the opposite, or inverse, of an exponential expression
    2. Taking the log of an exponential will 'undo' the exponent
    1. The definition says that y = bx and x = logby are equivalent equations
    2. So we can convert between the two forms and the expressions will be equal
    3. 'log' stands for 'logarithm'
    4. 'b' stands for 'base'
    5. 'x' is the exponent
    1. When we are told to 'evaluate' an expression, we want to find what it's equal to
    2. Since in our definition the log is equal to 'x', we can set our log equal to 'x' too since that's the value we're trying to find
    3. 'log' stands for 'logarithm'
    1. The base of the logarithm is the same as the base of the exponent
    1. 'log' stands for 'logarithm'
    2. 'b' stands for 'base'
    3. 'x' stands for 'exponent'
    1. The base of a logarithm is the small number down and to the right of 'log'
    2. 'log' stands for 'logarithm'
    1. Remember, the base of the logarithm is the 3
    2. 'x' is the number we're trying to find
    1. Right now our expression is in logarithmic form: log381 = x
    2. We want to rewrite it to be in exponential form to look like this: y = bx
    1. This gives us the equation in exponential form: 3x = 81
    2. In logarithmic form, 'y' is the number we're taking the log of
    3. log381 is in the form logby, so y = 81
    1. Now that we've gotten rid of that log, this equation is a little easier to work with
    1. 3x = 81 is asking us 'To what power do we raise 3 to get 81?'
    2. So let's guess some values for 'x' and see if any of them give us 81 when we plug them in
    3. 31 is 3
    4. 32 is 9
    5. 33 is 27
    6. 34 is 81
    1. So 4 is the solution to 3x = 81 and is our value for 'x'!
    1. 'log' stands for 'logarithm'
    2. Remember, bx = y and logb = x are the same thing
    3. So since 'x' equals 4 in 3x = 81, 'x' also equals 4 in log3 = x
    4. So when we evaluate log381, our answer is 4