LOGS EQUAL THE

The inverse of an exponential function is a logarithmic function. Logarithmic Function x = log a y read: “x equals log base a of y”

y = b x x = log b y These two equations are equivalent We can convert exponential equations to logarithmic equations and vice versa, using this:

Convert to exponential form 1) 2) 3)

Convert to logarithmic form 4) 5) 6)

Now that we can convert between the two forms we can simplify logarithmic expressions. Without a Calculator!

Simplify 1) log ) log ) log 4 2 4) log ? = 32 3 ? = 27 4 ? = 2 3 ? = 1 ? = 5 ? = 3 ? = 0.5 ? = 0 “What is the exponent of that gives you 32?” “ What is the exponent of 3 that gives you 27? ”

Evaluate

We can also use these two forms to help us solve for an inverse. The steps for finding an inverse are the same as before. Easy as 1, 2, 3… 1-Rewrite 2-Switch x and y 3-Solve for y

Example: Find the inverse

Now you try……..

An inverse you just have to know Ln and are inverses They undo each other 1. 2.

Example: Find the inverse

Now you try….. Find the inverse:

Essential Question: How do I graph & solve exponential and logarithmic functions? Daily Question: How do you expand and condense logs?

Change-of-Base Formula Let a, b, and x be positive real numbers such that a  1 and b  1.

Ex. 1 a) Evaluate using the change-of-base formula. Round to four decimal places.

Ex. 2 You can do the same problem using natural logarithms. a) Evaluate using the natural logarithms. Round to four decimal places.

Properties of Logarithms Product Property Quotient Property Power Property

Ex. 3 log 10 5x 3 y log log 10 y+ log 10 x 3 log log 10 y+ 3 log 10 x Expand.

Ex. 4 Expand.

Now you Try….. A.A. B.C. Expand each log 3 2x 6 y

Ex. 5 Condense. a) b)

Now you Try…..Condense each

Ex. 6 Use and to evaluate the logarithm.

Ex. 7 Use the properties of logarithms to verify that -ln ½ = ln 2 -ln ½ = -ln (2 -1 ) = -(-1) ln (2) = ln (2) =ln 2

Homework: Page 147  # 1 – 23 odd Page 157  # 1 – 25 odd