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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 2 32 2) log 3 27 3) log 4 2 4) log 3 1 2 ? = 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? ”
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 10 5+ log 10 y+ log 10 x 3 log 10 5+ 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
Properties of Logarithmic Functions
Essential Question: What are some of the similarities and differences between natural and common logarithms.
Let’s go over the homework!. DAILY CHECK Day 2 – Convert Logs to Exponentials, vice versa and Solve using one to one property.
8.4 Logarithms p. 486.
1 6.5 Properties of Logarithms In this section, we will study the following topics: Using the properties of logarithms to evaluate log expressions Using.
9.4 Properties of Logarithms. Since a logarithmic function is the inverse of an exponential function, the properties can be derived from the properties.
Properties of Logarithms
Copyright © Cengage Learning. All rights reserved. 3 Exponential and Logarithmic Functions.
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Properties of Logarithms. The Product Rule Let b, M, and N be positive real numbers with b 1. log b (MN) = log b M + log b N The logarithm of a product.
Warm - up.
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”
Essential Questions: 1. How do I solve exponential functions? 2. How do I solve logarmithic functions?
LAWS OF LOGARITHMS SECTION 5.6. Why do we need the Laws? To condense and expand logarithms: To Simplify!
Warm-up 1. Convert the following log & exponential equations 1. Convert the following log & exponential equations Log equationExponential Equation Log.
Logarithms the inverse of exponential functions. The logarithmic functions help us work easily with very large or very small numbers…. While calculators.
CONVERTING FROM ONE FORM TO ANOTHER EVALUATING PROPERTIES OF LOGS – EXPANDING AND CONDENSING Day 1:
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11.3 – Exponential and Logarithmic Equations. CHANGE OF BASE FORMULA Ex: Rewrite log 5 15 using the change of base formula.
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