# College Algebra : Solving Logarithmic Functions

## Example Questions

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### Example Question #1 : Solving Logarithmic Functions

Solve for :

Explanation:

To solve for , first convert both sides to the same base:

Now, with the same base, the exponents can be set equal to each other:

Solving for  gives:

### Example Question #2 : Solving Logarithmic Functions

Solve the equation:

Explanation:

### Example Question #3 : Solving Logarithmic Functions

Solve for .

Explanation:

Rewrite in exponential form:

Solve for x:

### Example Question #4 : Solving Logarithmic Functions

Solve the following equation:

Explanation:

For this problem it is helpful to remember that,

is equivalent to  because

Therefore we can set what is inside of the parentheses equal to each other and solve for  as follows:

### Example Question #5 : Solving Logarithmic Functions

Solve this logarithmic equation:

Explanation:

To solve this problem you must be familiar with the one-to-one logarithmic property.

if and only if x=y. This allows us to eliminate to logarithmic functions assuming they have the same base.

one-to-one property:

isolate x's to one side:

move constant:

### Example Question #6 : Solving Logarithmic Functions

Solve the equation:

No solution exists

Explanation:

Get all the terms with e on one side of the equation and constants on the other.

Apply the logarithmic function to both sides of the equation.

### Example Question #7 : Solving Logarithmic Functions

Solve the equation:

Explanation:

Recall the rules of logs to solve this problem.

First, when there is a coefficient in front of log, this is the same as log with the inside term raised to the outside coefficient.

Also, when logs of the same base are added together, that is the same as the two inside terms multiplied together.

In mathematical terms:

Thus our equation becomes,

To simplify further use the rule,

.

### Example Question #8 : Solving Logarithmic Functions

Solve the following equation for t

Explanation:

Solve the following equation for t

We can solve this equation by rewriting it as an exponential equation:

Next, take the fourth root to get:

We can check our work vie the following:

### Example Question #9 : Solving Logarithmic Functions

Solve for

It is impossible to isolate

Explanation:

The first step step is to carry out the inverse operation of the natural logarithm,

We can use the property  to simplify the left side of the equation to obtain,

Solve for

### Example Question #10 : Solving Logarithmic Functions

Solve this equation:

No solution.

None of these.

Explanation:

Note that this equation is a quadratic because

Factor:

Set each factor equal to 0 and solve:

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