Solving Exponential Equations - Math

Begin by recognizing that both sides of the equation have a root term of .

Using the power rule, we can set the exponents equal to each other.

Begin by recognizing that both sides of the equation have the same root term, .

We can use the power rule to combine exponents.

Set the exponents equal to each other.

is the only choice from those given that satisfies the equation. Substition of for gives:

Take the common logarithm of both sides and solve for :

, so can be rewritten as

One method: Take the natural logarithm of both sides and solve for :

Since , we can rewrite this equation by subsituting and applying the power rule:

This statement is identically false, which means that the original equation is identically false. There is no solution.

, so we can rewrite the equation as follows:

To find the x-intercepts, we set the numerator equal to zero and solve.

However, the square root of a number can be both positive and negative.

Therefore the roots will be

To find the y-intercepts, set and solve.

To find the x-intercepts, set the numerator equal to zero and solve.

We can simplify from here:

Now we need to rationalize. Because we have a square root on the bottom, we need to get rid of it. Since , we can multiply to get rid of the radical in the denominator.

Since we took a square root, remember that our answer can be either positive or negative, as a positive squared is positive and a negative squared is also positive.

To find the y-intercepts, set and solve.

To find the y-intercept, set and solve.

To find the x-intercepts, set the numerator equal to zero.

To find the y-intercept, set and solve.

The question gives us P (48,000) and asks us to find t (time). We can substitute for P and start to solve for t:

Now we have to isolate t by taking the natural log of both sides:

And since , t can easily be isolated:

Note: does not equal . You have to perform the log operation first before dividing.

To solve for in the equation

Factor out of the expression on the left of the equation:

Use the "difference of squares" technique to factor the parenthetical term on the left side of the equation.

Any variable that causes any one of the parenthetical terms to become will be a valid solution for the equation. becomes when is , and becomes when is , so the solutions are and .

Pull an out of the left side of the equation.

Use the difference of squares technique to factor the expression in parentheses.

Any number that causes one of the terms , , or to equal is a solution to the equation. These are , , and , respectively.

When dealing with exponential equations, we want to make sure the bases are the same. This way we can set-up an equation with the exponents.

With the same base, we can now write

Subtract on both sides.

Divide on both sides.