Logarithms

Converting Between Exponents and Logarithms

The answer is the exponent in a logarithm The answer of the exponent is the thing you take the log of.

Compound Interest Rate

Suppose that a principle $p is invested at the interest rate of r, compounded n times per year, then after t years - you will have this much money.

How do you use exponential functions

1. Population dynamics 2. Radioactive decay 3. Compound Interest in Investments 4. Newton's Law of Cooling 5. Richter Scale for earthquakes 6. Sound intensity - decibel levels 7. Learning curve

Logarithmic Graph Transformations

Adding or subtracting something outside the log does not change the vertical asymptote. Adding or subtracting something inside the log actually changes the vertical asymptote.

Using the log button on your calculator

log = log base 10 ln = log base e

Log base a of x (a can not be negative or the number 1)

what power do I need to raise the base "a" to in order to get the quantity that I'm taking the log of.

Exponential Graph Transformations

* 3 to the x compared to 3 to the x-1 shifts the graph one unit to the right. Inside addition (+n) moves to the left. Inside subtraction (-n) moves to the right. * Any constant outside the exponent shifts the whole graph either Down (-n) or Up (+n)

Word derivation - logarithm

It comes from the Greek words 'logos' and 'arithmes'. The second word means 'number'. The first word has a broad range of meanings, starting from 'word' going into 'thought' and 'reason' (thus 'logical'), and on to 'proportion'. Thus, a 'logarithm' is a 'proportion number'. I suppose the thinking was that logarithms help in the calculation of proportions: instead of multiplying a number by a fraction, you just add or subtract the logarithm of the fraction to the logarithm of the number. The logarithm actually predates the exponential function, one of the oddities of math history.

Euler's Constant

e = 2.7182818284590452353602874 - a transcendental, non repeating number like pi. It comes from geometry, interest on a loan, and continued fractions. The exponential function comes from Euler's constant.