Switch x and y = 1 / = = = 9

b > 0 and b = 1 V.A. at x = 0 Common Point at ( 1, 0 ) Domain: Range: b > 1: increase; 0 < b < 1: decrease b > 0 and b = 1 H.A. at y = 0 Common Point at ( 0, 1 ) Domain: Range: b > 1: increase; 0 < b < 1: decrease SWITCH X & Y CONCEPTS!

Negative, flip over the x-axis. 0 1, Vertical Stretch. Negative, flip over y-axis. 0 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 = 0x = 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 = 0 x = -2

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

Base Exponent WE MUST MEMORIZE THIS CONVERSION RULE!

Base e is the value …... Replace f(x) with y. Notice that there is no base number listed. Put in a 10. 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. Has to be +8 because the base must be positive. The exponential function is NOT isolated. Divide by 3. Covert to logarithmic. Should be written with ln.

The base of 2 on the log and inside the ( )’s cancel out. The base of e and the ln in the exponent cancel out. The 2 log base 2 can be condensed into one log. 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. 36 = 6*6 Convert to exponential. Convert to natural log and solve for x. 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 Convert the decimal to a fraction Product Rule Quotient Rule Power Rule b Power Rule

Change-of-Base formula is Conversion Check

We will use the Product, Quotient, and Power Rule to expand. Simplify, if possible. Product Rule Power Rule Breakdown the 8 = 2 3 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. Count the factors A log for each. Plus logs from the top. Minus logs from the bottom. Simplify individually with Power Rule. One more.

We will use the Product, Quotient, and Power Rule to condense. Simplify, if possible. We must 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 log 5 (5). 2 Move all coefficients back inside as powers. Simplify the insides, if possible. Optional. Plus logs to the top. Minus logs to the bottom. We have a log in every term! topbottom Simplify the inside…factor. Perfect Cube Diff. of Squares

One-to-one Property Conversion Rule We have a single log, convert to exponential.

Solve Not yet One-to-one Property. Power Rule to move coefficients. Now, One-to-one Property. Cancel logs. The Domain must be > 0…no negative 8 as a solution. Condense to one log on the left side. Use the Power and Product Rule. Now that there is one log, convert to exponential

Solve Condense to one log on the left side. Use the Product Rule. Now that there is one log, convert to exponential base exponent FOIL Set = 0. Factor Now check answers…-9 doesn’t work. -9 creates negative values inside the log. Condense to one log on the left side. Use the Quotient Rule. Factor and simplify the inside. Now that there is one log, convert to exponential

Now we isolate the exponential expression. Convert to a log. Our answer for x is considered an exact value. You may be asked to convert to a rounded decimal answer. Base Change Formula. This is factorable. Notice the first power is double that of the second power. Set = 0 and solve for x. 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. Not Possible Convert to a log.

UGLY! Bases are different. Authors way. The author wants you to take either the common log or natural log of both sides. 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. Power Rule Product &Quotient Rule Base Change

Mr. Fitz’s way. Pick a base and convert it to a log of that base. I will go 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.