Logarithmic equations

In this section we will refer to

as one of the inverse properties.

When the base is e the inverse property takes the form

We will also need to use the properties of logarithms

Let a be a positive number with If u and v are positive numbers and k is a real number then,

1.

2.

3.

Here are the three properties stated for the natural logarithm.

1.

2.

3.

Example 1

Solve

Solution .

Since a base 6 logarithm is involved make both sides exponents of 6.

Using the inverse property we get

Solving this, we have = 18.

General Strategy

In solving logarithmic equations our general approach will be to use properties of logs to get one log in the equation, make both sides an exponent of an appropriate base, use the inverse property, and solve the resulting equation.

Example 2

Solve

Solution .

Using property 2, the equation can be written as

Exponentiating with e as the base, we get

Using the inverse property, yields

Multiplying both sides by x-1, we get

=

Adding to both sides,

Thus,

and = 1.052395697

Since this value is in the domain of ln(x) and the domain of ln(x -1), it is a valid solution.

Confirm that this is a reasonable answer by graphing and y = 3 on the same coordinate system and looking at their intersection.

Example 3

Solve

Solution .

First isolate the ln(x) by adding 1 to each side and then dividing by 2.5

= 2

Exponentiating with e as the base, we get

Using the inverse property,

= 7.389056099

To check that this is reasonable by graphing and y = 4 on the same coordinate system.

Example 4

Solve

Solution .

Using property 1, this equation can be written as

Since this is the common log, we will make both sides exponents of 10.

Using the inverse property, we have

Putting this into standard form for a quadratic, we get

This quadratic has two solutions, = and Since this last solution is negative it is outside the domain of log(x) and so cannot be considered for the original equation. The solution is

which is approximately 2.701562119

Graphing y = log(x) + log(x + 1) and y = 1 confirms that this is a reasonable result.