# SAT II Math II : Mathematical Relationships

## Example Questions

### Example Question #23 : Exponents And Logarithms

Solve .

Explanation:

We can start by gathering all the constants to one side of the equation:

Next, we can multiply by  to change the signs:

Now we can rewrite the equation in exponential form:

And finally, we can solve algebraically:

### Example Question #24 : Exponents And Logarithms

Solve for :

Explanation:

In order to solve this problem, rewrite both sides of the equation in terms of raising  to an exponent.

Since, , we can write the following:

Since , we can write the following:

Now, we can solve for  with the following equation:

### Example Question #31 : Mathematical Relationships

Solve

No solutions

No solutions

Explanation:

The first thing we need to do is find a common base. However, because one of the bases has an  in it (an irrational number), and the other does not, it's going to be impossible to find a common base. Therefore, the question has no solution.

### Example Question #61 : Sat Subject Test In Math Ii

Solve

No solutions

Explanation:

First, we can simplify by canceling the logs, because their bases are the same:

Now we collect all the terms to one side of the equation:

Factoring the expression gives:

So our answers are:

### Example Question #62 : Sat Subject Test In Math Ii

Solve .

No solutions

Explanation:

Here, we can see that changing base isn't going to help.  However, if we remember that and number raised to the th power equals , our solution becomes very easy.

### Example Question #21 : Exponents And Logarithms

To the nearest hundredth, solve for .

None of these

None of these

Explanation:

Take the natural logarithm of both sides:

By the Logarithm of a Power Rule the above becomes

Solve for :

.

This is not among the choices given.

Define .

Evaluate .

Explanation:

### Example Question #1 : Absolute Value

Define .

Order from least to greatest:

Explanation:

, or, equivalently,

From least to greatest, the values are

### Example Question #61 : Sat Subject Test In Math Ii

Define an operation  as follows:

For all real numbers ,

Evaluate .

Undefined.

Explanation:

### Example Question #67 : Sat Subject Test In Math Ii

Define an operation  as follows:

For all real numbers ,

If , which is a possible value of ?