# College Algebra : Solving Exponential Functions

## Example Questions

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### Example Question #1 : Natural Log

Solve for .

Explanation:

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.

Explanation:

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.

No solution

Explanation:

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:

Explanation:

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:

Explanation:

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

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:

Case 1:

All real values of  are solutions:

Case 2:

Explanation:

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.

Explanation:

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:

Explanation:

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

### Example Question #2 : Solving Exponential Functions

Find the value of a:

Explanation:

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

Solve for .