Exponential and Logarithmic Functions - Pre-Calculus

First, you can begin to simplfy the numerator by converting all 3 expressions into base 2.

, which simplifies to

For the denominator, the same method applies. Convert the 25 into base 5, and when simplified becomes simply 5.

The final simplified answer becomes:

Let's work through this equation involving exponents one term at a time. The first term we see is , for which we can apply the following property:

So if we plug our values into the formula for the property, we get:

Because . Our next term is , for which we'll need the property:

Using the values for our term, we have:

The third term of the equation is , for which the quickest way to evaluate would be using the following property:

Using the values from our term, this gives us:

The next property we will need to consider for our fourth term is given below:

If we plug in the corresponding values from our term, we get:

Finally, our last term requires knowledge of the following simple property: Any number raised to the power of zero is 1. With this in mind, our last term becomes:

Rewriting the equation with all of the values we've just evaluated, we obtain our final answer:

Using the properties of exponents, we can either choose to subtract the exponents of the corresponding bases or rewrite the expression using negative exponents as such:

Here, we combine the terms with corresponding bases by adding the exponents together to get

Placing the x term (since it has a negative exponent) in the denominator will result in the correct answer. It can be shown that simply subtracting the exponents of corresponding bases will result in the same answer.

To remove the fractional exponents, raise both sides to the second power and simplify:

Now solve for :

To remove the rational exponent, cube both sides of the equation:

Now simplify both sides of the equation:

To solve this problem, recall that you can set exponents equal to eachother if they have the same base.

See below:

So, we have

Because both sides of this equation have a base of seven, we can set the exponents equal to eachother and solve for t.

We proceed as follows

Write as a fraction

The denominator of the fraction is a , so it becomes a square root.

Take the square root.

Raise to the power.

We begin by taking the natural log of the equation:

Simplifying the left side of the equation using the rules of logarithms gives:

We group the x terms to get:

We reincorporate the exponents into the logarithms and use the identity property of the natural log to obtain:

We combine the logarithms using the multiplication/sum rule to get:

We then solve for x:

We begin by factoring out the term to get:

This equation gives our first solution:

Then we check for more solutions:

Therefore our solution is

Remember the denominator of a rational exponent is equivalent to the index of a root.

This should simplify quite nicely.

When it gives us,

Recall that when considering rational exponents, the denominator of the fraction tells us the "root" of the expression.

Thus in this case we are taking the fifth root of .

The fifth root of is , because .

Thus, we have reduced our expression to .

What does an exponent of one-third mean? Consider our expression and raise it to the third power.

Simplifying, we get:

Thus, we are looking for a number that when cubed, we get . Thus, we are discussing the cube root of , or .

Simplify the constants:

Subtract the "x" exponents:

This is how the x moves to the denominator.

Finally subtract the "y" exponents:

When dividing two exponents with the same base we subtract the exponents:

Negative exponents are dealt with based on the rule

:

Subtract the "x" exponents and the "y" exponents vertically. Then add the exponents horizontally if they have the same base (subtract the "x" and subtract the "y" ones). Finally move the negative exponent to the denominator.

When an exponent is raised to the power of another exponent, just multiply the exponents together.

When an exponent is raised to the power of another exponent, just multiply the exponents together.

When an exponent is raised to the power of another exponent, just multiply the exponents together.

The derifintion of logarithm is:

In this problem,

Therefore,