Exponential and Logarithmic Functions - College Algebra

,

So,

This is a general formula that you should memorize. The inverse of is . You can use this formula to change an equation from a log function to an exponential function.

To solve , it is necessary to know the property of .

Since and the terms cancel due to inverse operations, the answer is what's left of the term.

The answer is:

Rewrite the following expression as an exponential expression:

Recall the following property of logs and exponents:

Can be rewritten in the following form:

So, taking the log we are given;

We can rewrite it in the form:

So b must be a really huge number!

Convert the following logarithmic equation to an exponential equation:

Recall the following:

This

Can be rewritten as

So, our given logarithm

Can be rewritten as

Fortunately we don't need to expand, because this woud be a very large number!

Convert the following logarithmic equation to an exponential equation.

To convert from logarithms to exponents, recall the following property:

Can be rewritten as:

So, starting with

,

We can get

To solve the following, you must "undo" the 5 with taking log based 5 of both sides. Thus,

The right hand side can be simplified further, as 125 is a power of 5. Thus,

Apply the Product of Powers Property to rewrite the second expression:

Distribute out:

Divide both sides by 5:

Take the natural logarithm of both sides (and note that you can use common logarithms as well):

Apply a property of logarithms:

Divide by and evaluate:

, so rewrite the expression at right as a power of 3 using the Power of a Power Property:

Set the exponents equal to each other and solve the resulting linear equation:

Distribute:

Subtract and 1 from both sides; we can do this simultaneously:

Divide by :

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:

Rewrite in exponential form:

Solve for x:

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:

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:

To solve this equation, remember log rules

.

This rule can be applied here so that

and

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.

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,

.

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:

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 ,

Note that this equation is a quadratic because

Factor:

Set each factor equal to 0 and solve: