Properties of Logarithms

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

• 
  

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:  $\log_73+\log_76$  

•  $\log_79$  

•  $\log_7\left(\frac{1}{2}\right)$  

•  $\log_718$  

•  $\log_7729$  

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

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

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

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

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

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