Logarithmic Theorems 2

The logs rules work "backwards", so you can simplify ("compress"?) log expressions. When they tell you to "simplify" a log expression, this usually means they will have given you lots of log terms, each containing a simple argument, and they want you to combine everything into one log with a complicated argument. "Simplifying" in this context usually means the opposite of "expanding".

• Simplify log₂(x) + log₂(y).

Since these logs have the same base, the addition outside can be turned into multiplication inside:

log₂(x) + log₂(y) = log₂(xy)

The answer is log₂(xy).

• Simplify log₃(4) – log₃(5).

Since these logs have the same base, the subtraction outside can be turned into division inside:

log₃(4) – log₃(5) = log₃(⁴/₅)

The answer is log₃(⁴/₅).

• Simplify 2log₃(x). Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

The multiplier out front can be taken inside as an exponent:

2log₃(x) = log₃(x²)

• Simplify 3log₂(x) – 4log₂(x + 3) + log₂(y).

I will get rid of the multipliers by moving them inside as powers:

3log₂(x) – 4log₂(x + 3) + log₂(y)
= log₂(x³) – log₂((x + 3)⁴) + log₂(y)

Then I'll put the added terms together, and convert the addition to multiplication:

log₂(x³) – log₂((x + 3)⁴) + log₂(y)
= log₂(x³) + log₂(y) – log₂((x + 3)⁴)
= log₂(x³y) – log₂((x + 3)⁴)

Then I'll account for the subtracted term by combining it inside with division:

Remember

1) Rearrange Terms - moving all negative to the end

2) Exponent Law - Coefficients become exponents (move to the top).

3) Multiplication Law - Addition become multiplication

4) Division Law - Subtraction becomes division

5) Fractional exponents become roots

An exponential equation is an equation that contains a variable in the exponent. We solved problems of this type in a previous chapter by putting the problem into the same base. Unfortunately, it is not always possible to do this. Take for example, the equation 2^x = 17. We cannot put this equation in the same base. So, how do we solve the problem? We use the change of base formula!! We can change any base to a different base any time we want. The most used bases are obviously base 10 and base e because they are the only bases that appear on your calculator!!
Change of base formula Log_b x = Log_a x/Log_a b
Pick a new base and the formula says it is equal to the log of the number in the new base divided by the log of the old base in the new base.