Logarithm (Log) Rules and Their Matching Exponents Rules

One of the problems that I had in high school math was trying to remember lists of rules without knowing where the rules came from. This could have been because the rules weren't explained, weren't explained enough for me, or were taught when I was absent and I had to learn it from a dry text book. (Note: I recall missing over a week of school with the flu one year, and as a result had problems with trigonometry identities for the longest time. They started to make sense in college during a Calculus class when I was staring at a Unit Circle regarding some other topic.)

If the rules for logarithms were explained in the context for the rules for exponents, as shown below, I might not have struggled with them as much, because whatever I didn't recall from memorization, I would've been able to work out. Again, it's possible that these rules were explained, but I wasn't in class and I had to get someone else's notes or read it from the book.

Exponent Rules	Logarithm Rules
(xa)(xb) = xa + b	log ((xa)(xb)) = log xa + log xb
xa / xb = xa - b	log (xa / xb) = log xa - log xb
(xa)b = xab	log (xa)b = b log xa
x0 = 1 (x =/= 0)	logb 1 = 0
x1 = x	logb b = 1 (b > 0)
xn =/= 0, when x =/= 0	logb 0 is undefined
xn > 0, when x > 0	logb x is undefined when x < 0

So the rules for expanding or condensing logarithms come from the corresponding rules for exponents. It's just that the notation may make it look more confusing than it actually is.

The Logarithm base switch rule says log_bc = 1/log_cb. The equivalent with exponents can be shown with this example:

2³ = 8 and 8^1/3 = 2

The exponents in those two case are reciprocals.

log₂8 = 3 and log₈2 = 1/3

So log₂8 = 1 / log₈2, because the logs are reciprocals.

The Logarithm base change rule says log_bx = log_cx / log_cb.

As I recall, the main reason to derive this rule was because, back in the dark ages when I was in high school, we used log tables, which were printed in the back of a text book, and based on 10 (and there was probably another page based of natural log tables based on e). If you could convert the base to 10, then you could use the table.

There would also be instances when you had terms in different bases that had to be converted before you work with them.

This is one example that shows how it works:

log₂8 = 3
log₄8 = 3/2
log₄2 = 1/2 log₄8 / log₄2 = (3/2) / (1/2) = 3

If you divide log₁₀8 / log₁₀2, you would also get 0.90308.../0.30102..., which is 3.000.

If you divide ln 8 / ln 2, you would also get 2.07944.../0.6931..., which is 3.000.

Edits made to formatting, and fixing the consistency of the variables. I hit "publish" a little too soon.