Logarithmic Properties and Change of Base

Properties of Logarithms

What is a logarithm?

What is a logarithm?

• A logarithm is another way to express an exponent

• A log function is the inverse of an exponential function

• The log of a number "X" to a certain base "a" is the same as saying what is the exponent that I need to raise the base to, to get the value "x"

• If it is not specified, the base is 10

Rewriting exponential expressions as log expressions

• Exponential 5² = 25

• Logarithm log₅ 25 = 2

• Exponential raises a base to a power and outputs a value

• Logarithm takes a value in a base and outputs an exponent

Rewriting exponential expressions as logarithmic expressions

• (base)^exponent = argument (value)

• log (_base)(value) = exponent

• Notice where each piece is in the exponential expression and the logarithmic expression

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Examples

• 1. exponential: 4⁵ = 1023

• logarithmic: log₄ (1023) = 5

• 2.exponential: 10³ = 1000

• logarithmic: log₁₀ (1000) = 3

• 3. exponential: 7⁶ = 117,649

• logarithmic: log ₇ (117649) = 6

Multiple Choice

Evaluate log₈ 8

Hint:

log_base(value) = exponent

base^exponent = value

• 8

• -1

• 0

• 1

Multiple Choice

Rewrite log_b(xⁿ)

Hint:

log_base(value) = exponent

base^exponent = value

• nlog_bx

• (log_bx)ⁿ

• xⁿlog_bx

• log_b(xⁿ)

Multiple Choice

Write log₆ 216 = 3 in exponential form.

Hint:

log_base(value) = exponent

base^exponent = value

• 3⁶=216

• 6³=216

• 216⁶=3

• 3²¹⁶=6

Multiple Choice

Write 2³= 8 in logarithmic form.

Hint:

log_base(value) = exponent

base^exponent = value

• log₂ 8 = 3

• log₂ 3 = 8

• log₃ 8 = 2

• log₃ 2 = 8

Multiple Choice

Rewrite 3⁴ = 81 in logarithmic form.

Hint:

log_base(value) = exponent

base^exponent = value

• log₃4 = 81

• log₈₁3 = 4

• log₃81 = 4

• log₄81 = 3

Logarithmic Rules

Logarithmic Rules: Product Rule

• log (a*b) = log(a) + log (b)

• related to the exponential rule (x)^a * (x)^b = x^(a+b)

• the log of a product equals the sum of the logs of the factors

• log (30) = log (5*6) = log (5) + log(6)

Multiple Choice

Simplify: $$\log_73+\log_76$$  

• $$\log_79$$  

• $$\log_7\left(\frac{1}{2}\right)$$  

• $$\log_7729$$  

•  $\log_7729$  

Multiple Choice

Condense this expression to a single logarithm.

Logarithmic Rules: Quotient Rules

• log (a/b) = log (a) - log(b)

• related to the exponential rule: x^a/x^b = x^a-b

• the log of a quotient = difference of the log of the numerator - log of denominator

• log (9/4) = log 9 - log 4

Multiple Choice

Simplify: $$\log_4\left(x+4\right)-\log_4\left(x-5\right)$$  

• $$\log_49$$  

• $$\log_4\left(2x-1\right)$$  

• $$\log_4\left(x^2-x-20\right)$$  

• $$\log_4\left(\frac{x+4}{x-5}\right)$$  

Multiple Choice

Condense this expression to a single logarithm.

Logarithmic Rules: Power Rule

• log (a)^x = x log (a)

• related to the exponent rule: ( x^m) ⁿ = x^(m*n)

• log of a number raised to a power is the power * log (number)

• log 5² = 2 log 5

Multiple Choice

$$5\log_{9\ }x$$  

Condense into a single logarithm. Simplify if possible.

• $$\log45x$$  

• $$\log_95x$$  

• $$\log_9x^5$$  

• $$\log_{ }14x$$  

Multiple Choice

$$3\log_45$$  

Condense into a single logarithm. Simplify if possible.

• $$\log_415$$  

• $$\log_4125$$  

• $$\log_43^5$$  

• $$\log_{ }60$$  

Compare exponent and log rules

You can also combine these rules.

Multiple Choice

$$\log_b\ \left(x^3\div yz^5\right)$$   $$\log_b\ \left(x^3\right)$$  Condense:   $$3\log_bx\ -\ \log_by\ -\ 5\log_bz$$
1. Rewrite as  $$3\log_bx\ -\ \left(\log_by\ +\ 5\log_by\right)$$ $$3\log_bx\ =\ \log_bx^3$$  

$$\log_by\ =\ \log_by$$  
$$5\log_bz\ =\ \log_bz^5$$  
 

• $$\log_b\ \frac{x}{yz^5}$$  

• $$\log_b\ \frac{x^3}{yz^{ }}$$  

• $$\log\frac{x}{yz^5}$$  

• $$\log_b\ \frac{x^3}{yz^5}$$  

Multiple Choice

Expand: $$\log_4\left(3x^2\right)$$  

Use exponent rule and product rule   $$2\log_43x\ =\ 2\ \left(\log_43\ +\ \log_4x\right)$$  

• $$2\log_43x$$  

• $$2\log_43\ +\ 2\log_4x$$  

• $$\log_43\ +\ 2\log_4x$$  
  

• $$2\log_43\ +\ \log_4x$$  

Multiple Choice

Condense: $$\log\left(5x+2\right)\ -\ \log3\ -\ \log x$$  Rewrite:  $$\log\ \left(5x\ +\ 2\right)\ -\ \left(\log_{\ }3\ +\ \log x\right)$$  Use product and quotient rules

$$\log\left(5x+2\right)\ \div\log\left(\ 3x\right)$$  

• $$\log\left(\frac{3x}{5x\ +\ 2}\right)$$  

• $$\log\left(\frac{5x\ +\ 2}{3x}\right)$$  

• $$\log\left(3x\left(5x+2\right)\right)$$  

• $$\log\left(\frac{5x\ +\ 2}{3\ -\ x}\right)$$