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Solving
Logarithmic Equations
Exponential
functions and logarithmic functions are closely related, they
are the inverse of each other. For an Exponential function  ,
the inverse function is defined by  .
We say  is
the log to the base b of x is the exponent to which b must be
raised in order to get x. Both
exponential and log are one-to-one functions, which have the
following properties:
Furthermore we know that  for
all real value x and  for
all positive real value y. We have two special identities.
Remember that a logarithm is an exponent and to solve a logarithmic
equation, we are frequently rewriting an equation in logarithmic
form to its equivalent exponential form. Namely,
 is
equivalent to 
Logarithmic functions obey certain rules:
Example
1: Solve 
Rewrite
the equation in exponential form and we have the answer immediately.
Example 2: Solve equation
for b: 
Sol: Rewrite the equation in exponential form
and remember that b must be greater than zero.
Example 3: Find 
Sol: Rewrite the equation in exponencial form.
Example
4: Solve 
Sol:
Example
5:
Solve 
Sol:
So the answer is
x = 5.
Example
6:
Solve 
Sol:
Example 7: Solve and leave the answer to two decimal places:
Sol: Rewrite the equation in logarithmic
form:
Note: Using change of base: 
Example 8: Solve the equation
for t and approximate t to 3 decimal places:
Sol:
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