Solving Exponential Functions - College Algebra

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 .

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:

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:

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,

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.

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,

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

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.

To solve for , set the two inside quantities equal to each other and solve.

Since the bases of the exponents are the same (4), the exponents have to equal each other for the exponential equation to be true.

In order to solve for x when it is in the exponent, we need to make the bases on either side of the equation look the same. In order to do this, factor the larger number until it looks like the smaller one. For this problem that looks as follows:

Factor the larger number:

Now substitute in the factored number:

Since the bases are now the same:

In order to solve for x when it is in the exponent, we need to make the bases on either side of the equation look the same. In order to do this with a fraction, factor the denominator then use the laws of exponents to eliminate the fraction. For this problem that looks as follows:

Manipulate the fraction:

Now substitute in the factored number:

Since the bases are now the same:

In order to solve for x when it is in the exponent, we need to make the bases on either side of the equation look the same. In order to do this, factor the larger number until it looks like the smaller one. For this problem that looks as follows:

Factor the larger number:

Now substitute in the factored number:

Since the bases are now the same:

In order to solve for x when it is in the exponent, we need to make the bases on either side of the equation look the same. In order to do this, factor the larger number until it looks like the smaller one. For this problem that looks as follows:

Factor the larger number:

Now substitute in the factored number:

Since the bases are now the same:

In order to solve for x when it is in the exponent, we need to make the bases on either side of the equation look the same. In order to do this, factor the larger number until it looks like the smaller one. For this problem that looks as follows:

Factor the larger number:

Now substitute in the factored number:

Since the bases are now the same:

In order to solve for x when it is in the exponent, we need to make the bases on either side of the equation look the same. In order to do this, factor the larger number in the denominator then manipulate the fraction until it looks like the left side of the equation. For this problem that looks as follows:

Factor the larger number and manipulate the fraction:

Now substitute in the factored number:

Since the bases are now the same:

In order to solve for x when it is in the exponent, we need to make the bases on either side of the equation look the same. In order to do this, factor the larger number until it looks like the smaller one. For this problem that looks as follows:

Factor the larger number:

Now substitute in the factored number:

Since the bases are now the same:

In order to solve for x when it is in the exponent, we need to make the bases on either side of the equation look the same. In order to do this, divide both sides by the coefficient in order to get the exponent by itself. Then factor the larger number until it looks like the smaller one. For this problem that looks as follows:

Divide both sides by 2:

Factor the larger number:

Now substitute in the factored number:

Since the bases are now the same:

In order to solve for x when it is in the exponent, we need to make the bases on either side of the equation look the same. In order to do this, divide both sides by the coefficient in order to get the exponent by itself. Then factor the larger number until it looks like the smaller one. For this problem that looks as follows:

Divide both sides by 3:

Factor the larger number:

Now substitute in the factored number:

Since the bases are now the same: