Exponential and Logarithmic Functions - Math

FOIL:

These are our possible solutions. However, we need to test them.

:

The equation becomes . This is true, so is a solution.

:

However, negative numbers do not have logarithms, so this equation is meaningless. is not a solution, and is the one and only solution. Since this is not one of our choices, the correct response is "The correct solution set is not included among the other choices."

Since and , it follows that and

Recall that by definition a logarithm is the inverse of the exponential function. Thus, our logarithm corresponds to the value of in the equation:

.

We know that and thus our answer is .

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:

Logs are exponential functions using base 10 and a property is that you can combine added logs by multiplying.

You cannot take the log of a negative number. x=-25 is extraneous.

Rewrite as a product that includes the number :

Then we can split up the logarithm using the Product Property of Logarithms:

Thus,

.

To solve this logarithm, we need to know how to read a logarithm. A logarithm is the inverse of an exponential function. If a exponential equation is

then its inverse function, or logarithm, is

Therefore, for this problem, in order to solve for , we simply need to solve

which is .

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:

This question is testing the definition of logs. is the same as .

In this case, can be rewritten as .

Taking square roots of both sides, you get . Since only the positive answer is one of the answer choices, is the correct answer.

The first step is to rewrite this equation in log form.

When rewriting an exponential function as a log we must remember that the form of an exponential is:

When this is rewritten in log form it is:

.

Therefore we have which when rewritten gives us,

.

Logarithms are another way of writing exponents. In the general case, really just means . We take the base of the logarithm (in our case, 2), raise it to whatever is on the other side of the equal sign (in our case, 4) and set that equal to what is inside the parentheses of the logarithm (in our case, x+6). Translating, we convert our original logarithm equation into . The left side of the equation yields 16, thus .

To solve this equation, remember log rules

.

This rule can be applied here so that

and

Because both sides have the same logarithmic base, both terms can be set equal to each other:

Now, evaluate the equation.

First, add x to both sides:

Add 15 to both sides:

Finally, divide by 6: .

In logarithmic expressions, is the same thing as .

Therefore, the equation can be rewritten as .

Both 8 and 128 are powers of 2, so the equation can then be rewritten as .

Since both sides have the same base, set .

Solve by dividing both sides of the equation by 3: .

Use the rule of Exponents of Logarithms to turn all the multipliers into exponents:

.

Simplify by applying the exponents: .

According to the law for adding logarithms, .

Therefore, multiply the 4 and 7.

.

Because both sides have the same base, .

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: