"The only way to be totally free is through education" - Jose Marti

Study Journal Notes

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Lesson 1:

• exponential function include at minimum a base and an exponent.

• if the base is between 0 and 1 (a positive fraction) then the function is a decaying exponential function

  • as these fractional bases increase, they get flatter (approaching the y = 1 line)​

• if the base is above 1 then the function is an exponential growth function

  • as these bases increase, they get steeper (going away from the y = 1 line)​

• transformation rules remain the same from our earlier unit (stretch, flip, translate)

• given a table of values, you can create an exponential function by identifying the base multiple (think of your power rules) and using trends of growth or decay to determine limitations on your base.

Lesson 2:

• the logarithmic function is the inverse of the exponential function (reflected about the line y=x)

• ***for logarithmic functions, the base cannot equal 1, the base must be greater than 0 and the input value (x) must be greater than 0.

• know the location of the asymptotes for basic exponents and basic logs

• logs are used to find the missing exponents

• logs and exponents can be transformed using the same rules as transformations for polynomials (from our first unit!)

Lesson 3:

• Know your Grade 10 exponent rules (these are at the top of page 8)

  • ***especially the rules for radicals and negative exponents​

• When in doubt, make the bases the same, that is the only way you can use the exponent rules to combing terms!

• If bases are the same and the terms are equal (see the top of page 9) then you can drop the bases and only inspect the exponents to solve for "x"

• If you have a variable base you can make the exponent equal to "1" (make sure whatever you do to the LHS of your equation you also do to the RHS of your equation). Your goal is to have your variable isolated on one side of the equal sign with no exponent.

Lesson 4:

• know the four properties of logarithms (inverse of exponents, can't have a negative input, same bases cancel, change the base by dividing with a common log base)

• Know the box in the middle of page 10 (power law, product law, quotient law)

• ***remember that the sign of the term directs the action of the term (negative terms will be divided, positive terms will be multiplied)

Lesson 5:

• There are two methods for solving exponential equations:

  • make the bases the same (we covered this in lesson 3)​

  • take the log of both sides (use this when the bases are different and uncommon/not factorable)

    • use logarithm laws to isolate for the variable (remember to use the law that moves the exponent to the coefficient spot for a log)​

Lesson 6:

• Know the methods for solving logarithms (top of page 14)

  • write all equations as logs with a common base including constants (no exponents)​

  • write each side of the equation as a single log using log properties (see lesson 4)

  • use log equivalencies to eliminate logs (shown on page 14)

• Once you have a single log expression, convert it to exponential form

• ***remember that you cannot have negative values for the input of a log function

Lesson 7:

• The principles of exponents and logarithms can be applied in a number of common scenarios - calculating compound interest for an investment, calculating the future value of a recurring investment, calculating the remaining balance owing on a loan that is accruing interest. 

  • Each of the above situations is modeled off of the compound interest equation on page 16, with slight adjustments made to account for continuous investments or continuous payments

• The doubling time and half-life formula will be provided for you on the 30-1 formula sheet. 

  • look for key words like "doubling time" to indicate the growth/decay factor (b)​

  • The period is the duration of the growth/decay factor

  • the time is the duration of observation

Lesson 8:

• Many of the worlds basic scales are logarithmic scales - Richter scale, Decibel scale, pH scale

  • these are base 10 log scales - each whole number increase in the scale corresponds to a 10x intensity increase​

• Solving problems for base-10 scales uses the magnitude formula found at the top of page 19 (note that this is a ratio compared to an exponent with base 10 - it is easily converted from log to exponent and vice-versa)

  • use the exponent rules (quotient rule) to examine differences in magnitude in each base-10 scenario.​