Chapter 6 - Logarithms - Concepts

Question 1

How do you find the inverse of something? (3)

Answer 1

• Flip x & y in coordinates
• Asymptotes flip between vertical & horizontal
• Domain and Range is flipped

Question 2

What is an exponential function? (5)
Equation, Domain, Range, Asymptotes, End Behavior

Answer 2

1. f(x) = a · bˣ
2. HA: always present; shifts with vertical translations
3. Domain: ℝ
4. Range: Never touches HA
   
   • a > 0 → (0, ∞)
   • a < 0 → (-∞, 0)
5. End Behavior:
   
   • Growth (b > 1, J shape): x→ ∞, f(x) → ∞ & x→-∞ f(x) → 0
   • Decay (0 < b < 1): x→ ∞, f(x) → 0 & x→-∞, f(x) → ∞

Question 3

What is a logarithmic function?
Domain, Range, Asymptotes, End Behavior

Answer 3

• Inverse function of exponential functions; Coordinates, D & R are flipped
• Approaches VA (x =__) instead of HA
• Domain: (vertical asymptote, infinity)
• Range: ℝ
• EB:
  As x→ ∞, f(x) → ∞
  As x→ -∞ f(x) → DNE
  As x → 0, f(x) → -∞

Question 4

What happens if there are transformations applied to logarithmic equation? (4)
f(x) = -log₃(-(x + 2)) + 4

Answer 4

f(x) = -log₃(-(x + 2)) + 4

• Reflection over x-axis: x→ ∞ → -∞ & x→ -2 → ∞
• Reflection over y-axis: D flips to (-∞, 0)
• Shift left 2: asymptote at x = -2, domain (-∞, -2)
• Shift up 4: graph moves upward, range unchanged (-∞, ∞)

Question 5

What is exponential form?

Answer 5

bˣ = C

Question 6

What is logarithmic form?

Answer 6

log₆(C) = x

Question 7

What is the relationship between exponential form and logarithmic form? (3)

Answer 7

• Exponential & logarithmic functions are inverses of each other
• Are able to convert between the two
• Subscript = base; parentheses = result; equals exponent

Examples:
- 3ˣ = 4 → log₃(4) = x
- log₂(8) = x → 2ˣ = 8 → x = 3

Question 8

How do you type in a logarithmic function in a calculator?

Answer 8

To choose base on calculator: Alpha, Window, 5

Question 9

What is natural logarithm? (3)

Answer 9

• Logarithm with a base of e (e ≈ 2.71828)
• ln(x) = logₑ(x)
• Can attach to both sides of an equation in exponential form and use power rule to solve for x

Question 10

What are common logarithms? (3)

Answer 10

• Logarithm with base 10
• No subscript present, can assume base is 10
• Can attach to both sides do an equation in exponential form and use power rule to solve for x

Question 11

What are three ways to solve for x in exponential form? (1/3/3)
6ˣ = 5

Answer 11

1. Define X: Convert to logarithmic form
   Ex: log₆5 = X
2. Use Common Logarithms (base 10)
   - Attach “log10” to both sides of equation
   - Use Power Rule (logbᵃ = c -> alogb = c)
   Ex: log6ˣ = log (5) ->xlog(6) = log(5) -> x = log(5)/log(6)
3. Use Natural Logarithms (base e)
   - Attach ln of both sides
   - Use Power Rule (lnᵃ = c -> alnb = c)
   Ex: ln6ˣ = ln5 -> xln(6) =ln(5) -> X = ln(5)/ln(6)

Question 12

What do you do if they give you a logarithmic function and ask you to evaluate? (3)
log(6)

Answer 12

• Set logarithmic form equal to X
• Convert to exponential form
• Find & drop common bases to solve for x

Question 13

What do you do if they give you an exponential function and ask you to solve for the inverse? (5)

Answer 13

• Substitute X & Y
• Simplify as much as possible to isolate X
• Should get in exponential form (bˣ = C)
• Convert to logarithmic form
• Isolate Y and substitute w/ f^-1(x)

Question 14

What is a list of all properties to solve logarithmic functions:

Answer 14

1. Conversion between exponential & logarithmic form
   bˣ = C <–> log₆(C) = x
2. Power Rule
   logaᵇ <–> bloga
3. Product Rule
   log (a ⋅ b) <–> loga + logb
4. Quotient Rule
   log (a/b) <–> loga - logb

Question 15

What is the power rule in logarithmic functions? (2)

Answer 15

1. Log of a number raised to a power can be rewritten as the power multiplied by the log of base number AND VICE VERSA
2. logaᵇ <–> bloga

Question 16

What is the product rule in logarithmic functions? (2)

Answer 16

1. Multiplying inside the log corresponds to addition outside AND VICE VERSA
2. log (a ⋅ b) <–> loga + logb

Question 17

What is the quotient rule in logarithmic functions? (2)

Answer 17

1. Division inside the log corresponds to the subtraction outside AND VICE VERSA
2. log (a/b) <–> loga - logb

Question 18

What is a sequence of steps you must follow in order to expand following logs? (4)

Answer 18

1. First, look for any fractions (apply quotient rule)
   log (a/b) –> loga - logb
2. Look for any products (apply product rule)
   log (a ⋅ b) –> loga + logb
3. Next, handle exponents/roots (handle power rule)
   logaᵇ –> bloga
4. Finally, distribute any negatives on the outside

Question 19

What is a sequence of steps you must follow in order to condense logs as a single log? (5)

Answer 19

1. If coefficient in front of log, move as power/radical (Power Rule)
   b ⋅ log(a) –> log(aᵇ)
2. If logs are being added, combine them into multiplication inside parentheses (Product Rule)
   log(a) + log(b) –> log(a ⋅ b)
3. If logs are being subtracted, combine them into division inside parentheses (Quotient Rule)
   log(a) - log(b) –> log(a / b)
4. Note: If coefficient is a fraction, apply root rather than power TO base in parentheses
5. Note: If there is a number in front of parentheses, apply as exponents to base in parentheses of log
   Sequence: Coefficients → Addition → Subtraction

Question 20

When you are given a logarithmic function (with a number GREATER THAN 1 IN PARANTHESES), and are asked to rewrite using a’s and b’s, how do you solve it? (4)
Ex: a = log2, b = log3, log (20)

Answer 20

• Split whatever is inside parentheses by any 2 factors
• Break up those factors equal to a/b/products of 10
• Apply product rule: Sum of numerous logs
• Convert to a, b and numbers
  Ex: log (20) -> log (2 x 10) -> log(2) + log (10) -> a + 1

Question 21

When you are given a logarithmic function (with a number LESS THAN 1 IN PARANTHESES), and are asked to rewrite using a’s and b’s, how do you solve it? (4)
Ex: a = log2, b = log3, log (0.003)

Answer 21

• Convert decimal into a fraction/quotient
• Convert numerator/denominator in terms of a/b/products of 10
• Apply quotient rule (log_-log_…)
• convert to a, b and numbers

Question 22

What are three types and therefore ways to solve logarithmic equations for x? (4)
5ˣ⁻¹ = 4
log₃ (x+2) = 2
log (3x + 5) = log (5)

Answer 22

• Given Expo Form & cannot get same base: Isolate x AMAP, then rewrite into log form
• Given Log Form = _: Condense AMAP, rewrite into exponential form
• Given Logs on both sides of equation: Drop logs w/ same base
• Must check/reject answer if does not align w/ domain (VA)

Question 23

How do you solve an equation in exponential form if bases cannot get dropped?
Ex: 4 (3)ˣ⁺² - 1 = 5

Answer 23

• Isolate X as much as possible
  Ex: 3ˣ⁺² = 3/2
• Convert to logarithmic form
  Ex: log₃(3/2) = X + 2
• Isolate x, either plug into calculator or leave in terms of log
  log₃(3/2) - 2 = X

Question 24

How do you solve an equation where both sides have logs in logarithmic form? (4)
2log₅(x+6) - log₅8 = log₅2

Answer 24

• Condense both sides as much as possible (preferably to single log)
  Ex: log₅(x+6)² - log₅8 = log₅2
  log₅ [(x+6)²/8] = log₅2
• Drop logs that have the SAME BASE
  Ex: [(x+6)²/8] = 2
• Solve for X
  Ex: √(x+6)² = √16 -> x = -6 (+/-) 4 -> x = -2, x = -10
• Choose/Reject X Values depending on domain
  Ex: Domain: (-6, INFINITY) -> REJECT: x = -10 -> Answer: x = -2

Question 25

How do you solve an equation in logarithmic form set equal to a number?

Answer 25

- Condense as much as possible (power/product/quotient) - Convert to Exponential Form (log_b (c) = x -> bˣ = c)

Question 26

How do you solve an equation in exponential form where both sides had x in exponent? Ex: 5ˣ⁺² = 4ˣ

Answer 26

- Convert to Log Form: Choose one side for base, number in (__) must include exponent w/ variable Ex: log₅(4ˣ) = x + 2 - Apply power rule by pushing exponent to front of equation Ex: xlog₅(4) = x + 2 - Rearrange the equation: Move "x" terms to one side & move other terms to opposite side Ex: xlog₅(4) - x = 2 - Factor out x from grouped terms & divide out to isolate x Ex: x (log₅(4) - 1) = 2 -> x = 2/(log₅(4) - 1)

Question 27

What is all stuff to keep in mind while solving logarithmic equations? (3)

Answer 27

- When given logs, try your best to condense as much as possible (use product, quotient & power rule) - When given expo form, try to isolate x as much as possible BEFORE converting to log form - Since domain is no longer all reals, must check if your answer aligns with domain (whatever makes parentheses 0 is VA, (VA, Infinity))