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There are three big rules to know when working with logs.

Product Rule: When adding logs with the same base. Multiply them.

logbM + logbN = logbM*N.

Quotient Rule: When subtracting logs with the same base. Divide them.

logbM - logbN = logbM/N

Power Rule: When there is an exponent in the log. You can move it behind.

logbM^N = NlogbM

LOGARITHMS

Common Logarithms: When you see a log without a base, it is assumed that it has a base of 10.

logx = log10x

Natural Logarithms: It is a log with a base of e.

loge = ln

Let's take a look at a problem that uses the Power Rule: 5^2x + 3 = 8 Step 1: Log both sides and because of the Power Rule, we can move 2x + 3 to the back. (2x + 3)log5 = log8 Step 2: To get rid of the log5, we can divide it on both sides. (2x+3)log5/log5 = log8/log5 --> 2x + 3 = log8/log5 Step 3: Plug in log8/log5 into your calculator to get an actual number. 2x + 3 = 1.29203 Step 4: Subtract 3 from both sides. 2x = -1.70797 Step 5: Finally divide both sides by 2 to get x by itself. x = -.854

EXAMPLE 1.

EXAMPLE 2.

Now we will do one with factoring: log2(x^2+4x) = log2(5) Step 1: Since both sides have log2, we can set whatever is inside equal to each other. x^2 + 4x = 5 Step 2: To factor, we need to set it equal to 0 and we can do this by subtracting 5. x^2 + 4x - 5 = 5 - 5 --> x^2 + 4x - 5 = 0 Step 3: Now we can use the X method to factor. What multiplies to be -5 and adds to be 4? 5 and -1 Step 4: Now rewrite the equation with the factors. (x + 5)(x - 1) = 0 Step 5: Finally set each factor to 0 and solve for x. x = -5, x = 1

Adding and subtracting rational expressions

When adding and subtracting rational expressions they might have different denominators.

Step one. Rewrite them so the have a common denominator

A good way to get a common denominator is to multiply them by each other

Step two. you would have to multiply by the original denominator

Step three. simplify all

step four. add them all across

Example 1.

5x/2x-3 + -4x^2/3x+1

Step 1. add common denominator

=5x(3x+1)/(2x-3)(3x+1)+(2x-3)(-4x^2)/(2x-3)(3x+1)

step 2. simplify

=15x^2 + 5x - 8x^3 + 12x^2/(2x-3)(3x+1)

step 3. fully simplify

= -8x^3+27x^2+5x/(2x-3)(3x+1)

Example 2.

9x^2 + 3/14x^2 - 9 - -3x^2 + 5/14x^2 - 9

step 1. keep denominator the same and subtract across

=9x^2 + 3 - (-3x^2 + 5)/14x^2 - 9

Step 2. Simplify

9x^2 + 3 +3x^2 + 5/14x^2 -9

step 3. Solve

=12X^2 - 2/14x^2 - 9

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