### All College Algebra Resources

## Example Questions

### Example Question #1 : Natural Log

Solve for .

**Possible Answers:**

**Correct answer:**

The first thing we notice about this problem is that is an exponent. This should be an immediate reminder: use logs!

The question is, which base should we choose for the log? We should use the natural log (log base e) because the right-hand side of the equation already has e as a base of an exponent. As you will see, things cancel out more nicely this way.

Take the natural log of both sides:

Rewrite the right-hand side of the equation using the product rule for logs:

Now rewrite the whole equation after bringing down those exponents.

is the same thing as , which equals 1.

Now we just divide by on both sides to isolate .

### Example Question #1 : Logarithms

Solve . Round to the nearest thousandth.

**Possible Answers:**

**Correct answer:**

The original equation is:

Subtract from both sides:

Divde both sides by :

Take the natural logarithm of both sides:

Divde both sides by and use a calculator to get:

### Example Question #2 : Logarithms

Solve for :

.

If necessary, round to the nearest tenth.

**Possible Answers:**

No solution

**Correct answer:**

Give both sides the same base, using *e*:

.

Because *e* and *ln* cancel each other out, .

Solve for x and round to the nearest tenth:

### Example Question #1 : Solving Exponential Functions

Solve the following for x:

**Possible Answers:**

**Correct answer:**

To solve this exponential function, you must first "undo" the e by taking the natural log of both sides. Thus,

Then, to isolate x, you must subtract two from both sides. Thus,

### Example Question #2 : Solving Exponential Functions

Solve the following for x:

**Possible Answers:**

**Correct answer:**

To solve, you must isolate x. The first two steps are to divide both sides by 2 and then take the natural log of both sides.

Finally, you must divide both sides by 4.

### Example Question #21 : Exponential And Logarithmic Functions

Give the solution set for the exponential equation shown below for the following cases:

**Case 1, **

**Case 2**

**Possible Answers:**

**Case 1: **

There are no solutions,

**Case 2: **

**Case 1: **

All real values of are solutions:

**Case 2: **

All real values of are solutions:

**Case 1: **

**Case 2: **

**Case 1: **

**Case 2: **

There is not enough information to find if we do not know the values of and b.

**Case 1: **

All real values of are solutions:

**Case 2: **

**Correct answer:**

**Case 1: **

All real values of are solutions:

**Case 2: **

**Case 1,**

This is true for all real values of , therefore,

**Case 2**

Take the natural logarithm of both sides (note that we could use any logarithm but it's convenient to just choose the natural logarithm).

Use the rule for pulling out exponents .

Note that if you were to divide out by , you would obtain which would imply which is not true for this case. We are solving for .

Expand with the distributive property,

Collect and isolate terms with onto one side of the equation,

Factor out (you could just factor out and leave the in front of the logarithms but it's easier to see the solution writing it this way).

What's remarkable about this solution is that it was obtained without specifying any value for or . The solution is true so as long as .

### Example Question #1 : Solving Exponential Functions

Solve for , round to the nearest hundredth.

**Possible Answers:**

**Correct answer:**

Solve for ,

Note the rules for logarithms,

** (1)**

** (2)**

** (3)**

** **

There is more than one way to get started, but notice that if we divide out by we can proceed as follows,

Take the natural log of both sides,

Use rule (3) to pull out the exponents,

Expand with the distributive property,

Collect terms with on onside and then factor out .

Divide out to solve for ,

** (4)**

** **

The numerator can be simplified using rules (2) and (3). Use rule (3) on the term in the numerator as follows,

So now the numerator can be written,

Bringing back the denominator in equation (4) gives,

### Example Question #21 : Exponential And Logarithmic Functions

Solve this equation:

**Possible Answers:**

**Correct answer:**

To solve , use the one-to-one property of exponential equations.

### Example Question #2 : Solving Exponential Functions

Find the value of a:

**Possible Answers:**

**Correct answer:**

For this problem, we can use the fact that the exponent and logarithm are inverses of each other. So if we raise both sides up as exponents:

The left side of the equation cancels and we get that

### Example Question #7 : Solving Exponential Functions

Solve for .

**Possible Answers:**

**Correct answer:**

To more easily solve for x, we need to try to express the right hand side of the equation in terms of a number of base 4. 256 is simply .

Rewriting the equation with this discovery yields

Since both bases are the same, we can set the exponents equal to each other and solve for x.

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