Solving logarithmic equations

Q:  Solve
log16x + log4x + log2x = 7

A:  It is easier (though not always necessary) to have all of the logs in the same base to proceed.  Since that is not the case, we need to use the change of base formula to put all of the logs into the same base.

Change of base formula says:
logab = logcb / logcwhere c can be any base of your choosing

What base should we go to?  Since 16, 4 and 2 can all be formed by powers of 2, let’s go to base 2:

log16x + log4x + log2x = 7

log2x / log216 + log2x / log24 + log2x / log22 = 7

Now, we can simplify:

log2x / 4 + log2x / 2 + log2x / 1 = 7

1/4 log2x + 1/2 log2x + log2x = 7

7/4 log2x = 7

log2x = 7 *(4/7)

log2x = 4  (remember, this reads:  the power you put on 2 to get x is 4)

x = 16

Solving Exponential Equations by Factoring

Q:  Solve for x:
22x + 2x+2 – 12 = 0

A:  The first thing we need to notice (from practice and experience) is that we can re-write this like so using our knowledge / rules of exponents:

(2x)2 + 22*2x – 12 = 0
Now simplify a little:

(2x)2 + 4*2x – 12 = 0

So, look at this in a new light.  What if we substitute each 2x with y?

(2x)2 + 4*2x – 12 = 0   turns into   y2 + 4*y – 12 = 0

(this isn’t necessary, but is helpful for visualization)

We see this is in the form of a quadratic and can be factored:

(y + 6)(y – 2) = 0

So, y = -6 or y = 2

Remember, y was a substitution for 2x. So, we really have:
2x = -6   or   2x = 2

Solve each equation separately.  Let’s start with:

2x = -6

Solve for x by taking the log of both sides (you can use the log of any base: log base 10, log base 2, natural log):

log(2x)= log(-6)
We can stop right here because you cannot take the log of a negative number. This equations yields no solutions.

So, the second equation:

2x = 2
log(2x) = log(2)

Logarithm rules say that the x exponent can come down as a multiplier like so:

x*log(2) = log(2)

Divide both sides by log(2) to get:

x = 1.

Solving Percent Problems using Proportions

Q:  How do you solve percent problems using proportions?

A:  Generally, there are 3 main parts to a percent problem:  the percent, the part and the total.

The set up is the same for all problems:

part/total = percent/100

(remember, percent means per 100, so it is always out of 100)

Example 1:  Finding the percent:

What percent is 75 of 85?

Set-up

75/85 = x/100

Cross multiply:

75*100 = 85*x

7500 = 85x

7500/85 = x

88.235% = x

So, 75 is 88.235% of the number 85.

Example 2:  Finding the “part”:

What is 76% of 50?

Set-up

x/50 = 76/100

Cross multiply:

100x = 76*50

100x = 3800

x = 3800/100

x = 38

So, 76% of 50 is 38.

Example 3:  Finding the “total”:

60 is 24% of what number?

Set-up

60/x = 24/100

Cross multiply:

24x = 60*100

24x = 6000

x = 6000/24

x = 250

So, 60 is 24% of the number 250.

Graphing Conic Sections

Q:  Identify the center, foci, vertices, and co vertices then graph the following conic section:
x2 + 9y2 = 36

A:  I am going to refer to the Conic Section guide that I had previously put together for students who I tutor:

Click to access conic_sections.pdf

This is a must have for those studying conic sections.  Click on it — it’s free 🙂

So, we look at our given equation:  x2 + 9y2 = 36 and recognize by its form that it is an ellipse.

To match the standard form of an ellipse, we need it equal, so divide both sides by 36 to get:

x2/36 + 9y2/36 = 36/36

Reduce to get:

x2/36 + y2/4 = 1

Refer to the Conic Section guide I linked to from above to follow along!

This is a horizontal ellipse (since the number under the x is larger than the number under the y)

The center is (0,0) since there are no values being subtracted from the x or the y.

The major axis:  a = sqrt(36) = 6  (in the x-direction)

The minor axis: b = sqrt(4) = 2  (in the y-direction)

The distance from the center to to foci can be found as follows:

c2 = a2 – b2

c2 = 36 – 4

c2 = 32

c = sqrt(32)

And, in simplest radical form, that is:

c = 4*sqrt(2)

So, the distance from the center to each foci (in the x-direction) is 4*sqrt(2)

Now, take all of the information from a, b and c found above to get the points:

Center: (0, 0)

Vertices:  (-6, 0) and (6, 0)

Co-Vertices:  (0, -2) and (0, 2)

Foci:  (-4*sqrt(2), 0) and (4*sqrt(2), 0)

Plot these points on a graph and then you’ve got it!

Solve the system of equations (conics)

Q:  Solve the system of equations (both of which are conic sections)
1:   x2 + y2 – 20x + 8y + 7 = 0
2:  9x2 + y2 + 4x + 8y + 7 = 0

A:  I am going to solve this by using the elimination method (since I see that I can cancel out all of the y’s)

I am going to multiply equation 1 by (-1) and then add it to equation 2:

2:  9x2 + y2 + 4x + 8y + 7 = 0

1: -x2 – y2 + 20x – 8y – 7 = 0   [after it has been multiplied by -1]

Now, add them together to get equation 3:

3:  8x2 +24x = 0

Now, equation 3 only has x’s so we can solve for x (by factoring):

(8x)(x + 3) = 0

So, x = 0 or -3

But, we aren’t done… Find the solution to this system of conics means that we are finding the point(s) of intersection — and it turns out there are two points of intersection since there were two solutions for x.  So, our points (solutions) are:

(0, ?)  and (-3, ?)

We need to find the y-coordinate of each solution separately.  We do this by plugging in the x value to either of the original equations (both will lead us to the same correct solution):

First x-value:  (0, ?)

x2 + y2 – 20x + 8y + 7 = 0

(0)2 + y2 – 20(0) + 8y + 7 = 0

y2  + 8y + 7 = 0

Solve for y by factoring:

(y + 7)(y + 1)=0

y = -7, -1

OH!  So this really means that there are more than two solutions…. When x = 0, we found two answers for y.

(0, -7) and (0, -1)

Second x-value:  (-3, ?)

x2 + y2 – 20x + 8y + 7 = 0

(-3)2 + y2 – 20(-3) + 8y + 7 = 0

9 + y2 +60 + 8y + 7 = 0

y2 + 8y + 76 = 0

Since this does not factor, we need to solve for y by using the quadratic formula.  I did not show my work here, but we end up with a negative under the square root, so there are no real solutions for when x = -3.

Therefore, there are two real solutions to this equation (2 points of intersection) and they are:

(0, -7) and (0, -1)

Here is a visual of what we are solving for….. We had the equations for the two graphed conic sections and we found the two points of intersection shown below:

Simplifying Radicals

Q:  Simplify:

A:  There are a few ways to do this, but first I will rationalize the denominator (that means get rid of the square root in the denominator) and then I will simplify the square root (simplest radical form):

Step 1:  Rationalize the denominator:

Multiply the top and bottom by sqrt(75):

Now, simplify algebraically:

Simplify the denominator to get 150… for some reason, I can’t make my picture keep it in a fully fraction with 150 in the bottom, so the image shows the 150 pulled out like so (either way is fine):

Now reduce the 15 and the 150 to get:

The denominator has been rationalized (no square roots in the denominator) and the fraction has been reduced.

Step 2:  Put the numerator into simplest radical form:

To go to simplest radical form, you want to see what “perfect squares” divide into the square root.  So, what perfect squares divide into 75?  25 is a perfect square that divides into 75:  3*25=75.

So, we can break up the square root like so (again, the square roots can be on top of the fraction, just having image problems):

Now, what is the square root of 25?  That’s 5, so simplify:

And, now reduce the fraction part of the 5 and the 10 in the denominator to get:

You are done!