Note:

If you would like an in-depth review of logarithms, the rules of logarithms, logarithmic functions and logarithmic equations, click on logarithmic functions.

Solve for x in the following equation.

**Problem 8.4d:**

and The approximate answers are 1.2478358458, 4.75216415412, and

**Solution:**

The above equation is valid only if the expression
The expression
is valid when
and Another way of saying this is that the
domain is the set of real numbers where

Note: If, when solving the above problem, you simplify the logarithmic
equation
to
you will lose half of your solutions.
Why?

Recall that
is valid for all
real numbers that are not equal to 1 or 5. However,
is valid for the set of real numbers
This means that
only over the set of real numbers

So be careful when you simplify a logarithmic equation before solving it.

Convert the logarithmic equation
to an equivalent exponential equation with base 0.982.

These answers may or may not be the solutions. You must check them
numerically or graphically.

**
Numerical Check:**

- Check the answer
by substituting
5.22034249092 in the original equation for
x. If the left side of the equation equals the right side of the equation
after the substitution, you have found the correct answer.

Left Side:

Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 5.22034249092 for x, then is a solution.

- Check the answer
by substituting
4.75216415413 in the original equation for
x. If the left side of the equation equals the right side of the equation
after the substitution, you have found the correct answer.

Left Side:

Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 4.75216415413 for x, then is a solution.

- Check the answer
by substituting
0.779657509082 in the original equation
for x. If the left side of the equation equals the right side of the
equation after the substitution, you have found the correct answer.

Left Side:

Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 0.779657509082 for x, then is a solution.

- Check the answer
by substituting
1.24783584587 in the original equation for
x. If the left side of the equation equals the right side of the equation
after the substitution, you have found the correct answer.

Left Side:

Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 1.24783584587 for x, then is a solution.

**
Graphical Check:**

You can also check your answer by graphing
(formed by subtracting the right side of the
original equation from the left side). Look to see where the graph crosses
the x-axis; that will be the real solution. Note that the graph crosses the
x-axis at four places:
,
and
5.22034249092. This means that
,
and
5.22034249092 are the real solutions.

If you have trouble graphing the function
,
graph the equivalent function
.

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