Common and Natural Logarithms

Common Logarithms A common logarithm has a base of 10. If there is no base given explicitly, it is common. You can easily find common logs of powers of ten. You can use your calculator to evaluate common logs.

Change of Base Formula Allows us to convert to a different base. If a, b, and n are positive numbers and neither a nor b is 1, then the following equation is true.

Natural Logarithms A natural logarithm has a base of e.

Natural Logs and “e” Start by graphing y=e x The function y=e x has an inverse called the Natural Logarithmic Function. Y=ln x

What do you notice about the graphs of y=e x and y=ln x? y=e x and y=ln x are inverses of each other! We can use the natural log to “undo” the function y= e x (and vice versa).

All the rules still apply You can use your product, power and quotient rules for natural logs just like you do for regular logs Let’s try one:

The mathematical constant e is the unique real number such that the value of the derivative (the slope of the tangent line) of the function f(x) = e x at the point x = 0 is exactly 1. The function e x so defined is called the exponential function. The inverse of the exponential function is the natural logarithm, or logarithm with base e. The number e is also commonly defined as the base of the natural logarithm (using an integral to define that latter in calculus), as the limit of a certain sequence, or as the sum of a certain series. The number e is one of the most important numbers in mathematics, alongside the additive and multiplicative identities 0 and 1, the constant π, and the imaginary number i.π e is irrational, and as such its value cannot be given exactly as a finite or eventually repeating decimal. The numerical value of e truncated to 20 decimal places is: –

Natural Logarithms A natural logarithm has a base of e. We write natural logarithms as ln. –In other words, log e x = ln x. If ln e = x…

Examples of evaluating expressions Change of base formula examples

Solving with base “e” 2. Divide both sides by 7 3. Take the natural log of both sides. 4. Simplify. 1. Subtract 2.5 from both sides 5. Divide both sides by 2 x = Calculator

Another Example: Solving with base “e” 1. Take the natural log of both sides. 2. Simplify. 3. Subtract 1 from both sides x = Calculator

Solving a natural log problem 2. Use a calculator 3. Simplify. 1. Rewrite in exponential form To “undo” a natural log, we use “e”

Another Example: Solving a natural log problem 1. Rewrite in exponential form. 2. Calculator. 3. Take the square root of each time 3x+5 = 7.39 or Calculator X=0.797 or Simplify

Let’s try some

Going back to our continuously compounding interest problems... A $20000 investment appreciates 10% each year. How long until the stock is worth $50000? Remember our base formula is A = Pe rt... We now have the ability to solve for t A = $50,000 (how much the car will be worth after the depreciation) P = $20,000 (initial value) r = 0.10 t = time From what we have learned, try solving for time

Going back to our continuously compounding interest problems... $20000 depreciates 10% each year. How long until the car is worth $5000? A = $50,000 (how much the car will be worth after the depreciation) P = $20,000 (initial value) r = 0.10 t = time