Math 3 Keeper 27 Logarithms Essential Question – How is a log function related to an exponential function? You use log functions to solve exponential problems; they are inverses of each other.

When will I use this? Human memory Intensity of sound (decibels) Finance Richter scale

Evaluating Log Expressions We know 22 = 4 and 23 = 8 But for what value of y does 2y = 6? Because 22<6<23 you would expect the answer to be between 2 & 3. To answer this question exactly, mathematicians defined logarithms. Logarithms are the INVERSE of exponential functions.

Definition of Logarithm to base b Let b & x be positive numbers, b ≠ 1. logby = x iff bx = y This expression is read “log base b of y” The definition tell you that the equations logby = x and bx = y are equivalent.

Example 1: Rewrite the equation in exponential form Log form Exp. form a) log232= 5 b) log51 = 0 c)log101 = 1 d) Log1/2 2 = -1 25 = 32 50 = 1 101 = 1 (1/2)-1 = 2

YOUR TURN!! Log form Exp. form e) log39= 2 f) log81 = 0 g)log5(/25)=-2 32 = 9 80 = 1 5-2 = 1/25

Rewriting forms: To evaluate log3 9 = x ask yourself… “3 to what power is 9?” 3x = 9 → 32 = 9 so…… log39 = 2

Example 2: Evaluate the expression without a calculator log381 b) log50.04 c) log5125 4 -2 3

4 4x = 256 4x = 64 (1/4)x = 256 2x = (1/32) d) log4256 e) log464 YOUR TURN!! 4 4x = 256 4x = 64 (1/4)x = 256 2x = (1/32) d) log4256 e) log464 f) log1/4256 g) log2(1/32) 3 -4 -5

You should learn the following general forms!!! Log b 1 = 0 because b0 = 1 Log b b = 1 because b1 = b Log b bx = x because bx = bx

Natural logarithms log e x = ln x The natural log is the inverse of the natural base, e. ln means log base e

Common logarithms log 10 x = log x Understood base 10 if nothing is there.

Common logs and natural logs with a calculator log10 button ln button

Example 3: Use a calculator to evaluate the expression Example 3: Use a calculator to evaluate the expression. Round answer to 3 decimal places. log 5 b) ln 0.1 c) log 7 d) ln 0.25 0.6989= 0.700 -2.303 0.845 -1.386

INVERSE PROPERTIES g(x) = log b x is the inverse of the exponential function f(x) = bx f(g(x)) = blogbx = x g(f(x)) = logbbx = x *Exponential and log functions are inverses and “undo” each other

Example 4: Using inverses→ Simplify the expression. 2 a) 10log2 = b) log39x = c) 10logx = d) log5125x = log332x= 2x log3(32)x = x 3x log5(53)x = log553x =

Finding Inverses Find the inverse of y = log3x By definition of logarithm, the inverse is y=3x OR write it in exponential form and switch the x & y! 3y = x → 3x = y

Example 5: Find the inverse of... a) y = ln (x +1) X = ln (y + 1) Switch the x & y x = loge(y + 1) Write in log form ex = y + 1 Write in exp form ex – 1 = y Solve for y y = ex – 1 Final Answer

Example 5: Find the inverse of... b) y = log8x 8y = x Switch x & y 8x = y Solve for y y = 8x Final Answer

Example 5: Find the inverse of... y = ex + 3 y = (2/5)x y = ex + 10 c) y = ln (x - 3) d) y = log2/5x e) Y = ln (x–10)