# Warm-up Solve: log3(x+3) + log32 = 2 log32(x+3) = 2 log3 2x + 6 = 2

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Warm-up Solve: log3(x+3) + log32 = 2 log32(x+3) = 2 log3 2x + 6 = 2

Exponential and Logarithmic Equations Section 3-4

I can solve equations with exponents using logarithms
Objectives I can solve equations with exponents using logarithms

horizontal asymptote y = 0 y = log2 x
Graph f (x) = log2 x Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x. x y y = 2x y = x 8 3 4 2 1 –1 –2 2x x horizontal asymptote y = 0 y = log2 x x-intercept (1, 0) vertical asymptote x = 0 Graph f(x) = log2 x

Natural Logarithmic Function
y x 5 –5 y = ln x The function defined by f(x) = loge x = ln x (x  0, e ) is called the natural logarithm function. y = ln x is equivalent to e y = x Use a calculator to evaluate: ln 3, ln –2, ln 100 Function Value Keystrokes Display ln 3 LN 3 ENTER ln –2 LN –2 ENTER ERROR ln 100 LN 100 ENTER Natural Logarithmic Function

Log Functions Overview
Log Function Base loga x a log x 10 loge x e ln x e

Using Logarithms to Solve Exponents
So far we have solved exponents using the principle of getting the same base, then setting the exponents equal. There are many times that we cannot get the same base, so we need to solve a different method.

Old Problems vs New Solve for x 2(x+1) = 8 2(x+1) = 23 x+1 = 3 x =2
If this problem we cannot get the same base To work this new type problem, we will use logarithms

Rule for Logarithms Logarithms can be applied to equations.
In any equation, if I do something to one side, I must do the same thing to the other side to keep equality. In these problems, we will take the Common Log or Natural log of both sides of each equation, then use the Power Property

Solve for x 8(2x-5) = 5(x+1) log 8(2x-5) = log 5(x+1)

Natural log vs ex Use your calculator and determine the following:
Ln e1 = Ln e2 = Ln e3 =

Example 2 e3x = 20 ln e3x = ln 20 3x = ln 20 3x = x = .9986

Example 3 3 + ln x = -8 ln x = -11 eln x = e-11 x = 1.67 x 10-5

Homework WS 6-4

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