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Switch x and y. 3 -1 = 1 / 3. 3 0 = 1. 3 1 = 3. 3 2 = 9. SWITCH X & Y CONCEPTS!. b > 0 and b = 1. b > 0 and b = 1. H.A. at y = 0. V.A. at x = 0. Common Point at ( 0, 1 ). Common Point at ( 1, 0 ). Domain: Range:. Domain: Range:.
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Switch x and y. 3-1=1/3 30= 1 31= 3 32= 9
SWITCH X & Y CONCEPTS! b > 0 and b= 1 b > 0 and b= 1 H.A. at y = 0 V.A. at x = 0 Common Point at( 0, 1 ) Common Point at( 1, 0 ) Domain: Range: Domain: Range: b > 1: increase; 0 < b < 1: decrease b > 1: increase; 0 < b < 1: decrease
logarithmic function, Negative, flip over the x-axis. 0 < a < 1, Vertical Shrink and a > 1, Vertical Stretch. Negative, flip over y-axis. 0 < b < 1, Horizontal Stretch and b > 1, Horizontal Shrink. Solve for x. This is the Horizontal shift left or right. This is the Vertical shift up or down.
Translate Graph The negative will flip the graph over the x-axis. x = 0
Translate Graph The minus 2 is inside the function and solve for x. x = +2, shift to the right 2 units. x = 0 x = 2
Translate Graph The minus 5 is outside the function and shift down 5 units. x = 0
Translate Graph The negative on the x will flip the graph over the y-axis and solve 1 – x = 0 to determine how we shift horizontally. x = 1 x = 0 x = 1
Translate Graph The plus 2 is inside the function and solve for x. x = -2, shift to the left 2 units. The minus 1 will shift down 1 unit. x = -2 x = 0
Translate Graph The negative on the x will flip over the y-axis. The negative in front of the log will flip the graph over the x-axis. x = 0
Exponent Base WE MUST MEMORIZE THIS CONVERSION RULE!
Base e is the value 2.718281828…... Notice that there is no base number listed. Put in a 10. Replace f(x) with y. Switch x and y. Solve for y. Convert to exponential. Convert to logarithm.
Isolate the log function or exponential function. The log function is isolated. Covert to exponential. The exponential function is NOT isolated. Divide by 3. Covert to logarithmic. Has to be +8 because the base must be positive. Should be written with ln.
The base of e and the ln in the exponent cancel out. The 2 log base 2 can be condensed into one log. The base of 2 on the log and inside the ( )’s cancel out. 16 = 2*2*2*2
Set the exponent = x. Use the base change formula for both logs. The 2 log base 6 can be condensed into one log. Convert to exponential. Convert to natural log and solve for x. 36 = 6*6 Place the value of x back as an exponent on e.
The goal here is to factor the argument of the inside the log function to a 2, 3, or 5. Convert the decimal to a fraction. Quotient Rule Product Rule 3 3 3 3 4 2 2 Power Rule 5 2 2 3 5 b Power Rule
Change-of-Base formula is Conversion Check
We will use the Product, Quotient, and Power Rule to expand. Simplify, if possible. 2 Count the factors. 1 Product Rule A log for each. 3 4 Power Rule Minus logs from the bottom. Plus logs from the top. Simplify individually with Power Rule. Breakdown the 8 = 23 One more. Notice that there is a log for every factor. This is true whether the factors are top or bottom. Remember that the factors from the bottom are always minus the log.
We will use the Product, Quotient, and Power Rule to condense. Simplify, if possible. 2 top bottom We must have a log in every term! We have a log in every term! The 2 is the answer from a simplified log. This means that is was the exponent on the base inside of a log5(5). Perfect Cube Diff. of Squares Move all coefficients back inside as powers. Simplify the inside…factor. Simplify the insides, if possible. Optional. Plus logs to the top. Minus logs to the bottom.
Conversion Rule One-to-one Property We have a single log, convert to exponential.
Solve Condense to one log on the left side. Use the Power and Product Rule. Not yet One-to-one Property. Power Rule to move coefficients. Now, One-to-one Property. Cancel logs. Now that there is one log, convert to exponential The Domain must be > 0…no negative 8 as a solution.
Solve Condense to one log on the left side. Use the Product Rule. Condense to one log on the left side. Use the Quotient Rule. exponent base Now that there is one log, convert to exponential Factor and simplify the inside. FOIL Set = 0. Factor Now that there is one log, convert to exponential Now check answers…-9 doesn’t work. -9 creates negative values inside the log.
Now we isolate the exponential expression. This is factorable. Notice the first power is double that of the second power. Convert to a log. Set = 0 and solve for x. Our answer for x is considered an exact value. You may be asked to convert to a rounded decimal answer. Base Change Formula. Convert to logs for both. No negatives in log functions or exponentials can’t = negatives
This is NOT factorable. Notice the first power is NOT double that of the second power. FORCE IT! AARRRRGGGGG! Use your exponential rules. When you multiply like bases, we add the exponents. So manipulate the middle term. Set = 0 and solve for x. Convert to a log. Not Possible
UGLY! Bases are different. Authors way. Power Rule The author wants you to take either the common log or natural log of both sides. Product &Quotient Rule Power Rule Dist. Prop. Move all the terms with an x to the left side. Base Change Non- x terms to the right side. Factor out x as GCF. Isolate x.
Mr. Fitz’s way. Pick a base and convert it to a log of that base. I will go with base 5. Power Rule Dist. Prop. Move all the terms with an x to the left side. Non- x terms to the right side. Factor out x as GCF. Isolate x.
