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Solving a Quadratic Equation

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Exponential and Logarithmic Equations

Part 1: Solving Exponential Equations

We use the inverse properties to solve both exponential and logarithmic equations:

In particular, to solve exponential equations that don't have the same base on both sides of the equal sign, take logs of both sides. That is, use

If the base is not base 10, then use

Question 5: Solve

Since the bases are the same we can set the exponents equal to each other.

Question 17: Solve

Here the bases are different so let's take logs of both sides.

Bring the exponents down in front of the logs.

Divide both sides by the constant

This is the exact solution.
The approximate solution is

Question 43: Solve:

Multiply both sides by 2 to eliminate the fraction.

We can write this as

Multiply both sides by to eliminate the fraction.

This is like a quadratic equation! Use the quadratic formula:

Take ln of both sides

Question 49: Use MathCAD to solve graphically and analytically:
Rewrite this as
And graph each side on the same axis.

Notice that there is a solution close to x =1. If we zoom in we can get a closer approximation: x = 0.96

MathCAD won't solve it if you use Symbolics-Variable-Solve. But if you use the Given-Find block. It will solve it numerically:

Part 2: Solving Logarithmic Equations

To solve logarithmic equations, use the logarithmic identities to transform the equation into either one of the following.

1.
2.

In case 1., use the identity and take logs of both sides to rewrite the equation with without logs.
In case 2., use the same identity to transform the equation into the form

Question 30: Solve:

This is a quadratic equation
Solution is -10, -5
I can't use -10 because it would give me a negative argument for the log.
The only solution is -5

Question 25: Solve:

Now raise both sides as exponents of 2

Solution is:

Question 58: After a race, a runner's pulse rate R in beats per minute decreases according to the function

where t is measured in minutes.

a) Find the runner's pulse at the end of the race and also 1 minute after the end of the race.

b) How long, to the nearest minute, after the end of the race will the runner's pulse be at 80 beats per minute?

a) beats per minute at the end of the race
beats per minute

b) and solve for t.

Solution is:
About 6 minutes after the end of the race.

Question 69: The velocity v of an object t seconds after it has been dropped from a height above the surface of the earth is given by the equation v = 32t feet per second, assuming no air resistance. If we assume that air resistance is proportional to the square of the velocity, then the velocity after t seconds is given by

a) In how many seconds will the velocity be 50 feet per second?

b) Determine the horizontal asymptote for the graph of this function.

c) Write a sentence that describes the meaning of the horizontal asymptote in the context of this problem.

a) We need to solve

Multiply both sides by the denominator .

b) Change the function to

Now as t gets large without bound, the two little fractions disappear. So the horizontal asymptote is y = 100. See the graph below.

c) The maximum (terminal) velocity of the object is 100 feet per second