# SAT II Math II : Mathematical Relationships

## Example Questions

### Example Question #3 : Exponents And Logarithms

Solve for :

Give your answer to the nearest hundredth.

The equation has no solution.

Explanation:

Take the common logarithm of both sides and solve for :

### Example Question #4 : Exponents And Logarithms

Solve for :

Give your answer to the nearest hundredth.

The equation has no solution.

Explanation:

Take the natural logarithm of both sides and solve for :

### Example Question #5 : Exponents And Logarithms

To the nearest hundredth, solve for :

Explanation:

Take the common logarithm of both sides, then solve the resulting linear equation.

### Example Question #6 : Exponents And Logarithms

To the nearest hundredth, solve for :

The equation has no solution.

Explanation:

Take the common logarithm of both sides, then solve the resulting linear equation.

### Example Question #7 : Exponents And Logarithms

Solve for :

Explanation:

Take the common logarithm of both sides:

### Example Question #8 : Exponents And Logarithms

Solve for :

Explanation:

The base of the common logarithm is 10, so

The sum of three logarithms is the logarithm of the product of the three powers, so:

Therefore,

### Example Question #9 : Exponents And Logarithms

To the nearest hundredth, solve for :

Explanation:

### Example Question #10 : Exponents And Logarithms

Solve for :

The equation has no solution

Explanation:

The base of the common logarithm is 10, so

The sum of logarithms is the logarithm of the product of the three powers, and the difference of logarithms is the logarithm of the quotient of their powers. Therefore,

### Example Question #11 : Exponents And Logarithms

Give the set of real solutions to the equation

(round to the nearest hundredth, if applicable)

The equation has no solution.

Explanation:

Using the Product of Powers Rule, then the Power of a Power Rule, rewrite the first two terms strategically:

Substitute  for ; the equation becomes

Factor this as

by finding two integers whose product is 3 and whose sum is . Through some trial and error we find , so we can write

By the Zero Product Rule, one of these two factors must be equal to 0.

If , then .

Substituting  back for :

.

If , then .

Substituting  back for :

Both can be confirmed to be solutions by substitution.

### Example Question #12 : Exponents And Logarithms

Give the solution set:

The equation has no solution.

Explanation:

Rewrite by taking advantage of the Product of Powers Property and the Power of a Power Property:

Substitute  for ; the resulting equation is the quadratic equation

which can be written in standard form by subtracting  from both sides:

The trinomial can be factored by the  method, Look for two integers with sum  and product ; by trial and error, we find they are , so the equation can be rewritten and solved by grouping:

By the Zero Product Property, one of these factors must be equal to 0.

Either

Substituting  back for :

Or:

Substituting  back for :

The solution set, as can be confirmed by substituting in the equation, is .