Logarithms and Exponentials

Logarithms and Exponential Functions

Logarithms

Learning Objective

• understand the relationship between logarithms and indices.

• able to use the laws of logarithms to any base (excluding change of base)

Logarithms to base 10

• also called as common logarithms

•  $y=10^x\ \ \ \leftrightarrow\ \ \ x=\log_{10}y$  

•  $y=10^x\ \ \ \leftrightarrow\ x=\log y$  

Rules

•  $\log x+\log y=\log xy$  

•  $\log x-\log y=\log\ \frac{x}{y}$  

•  $\log y^m=m\log y$  

•  $\log_xy=\frac{\log y}{\log x}$  

Examples

•  $\log100=\log10^2=2\log10=2$  

•  $\log0.01=\log10^{-2}=-2.\log10=-2$  

•  $\log2+\log3-\log6=\log\left(\frac{2.3}{6}\right)=\log1=0$  

Multiple Choice

log 10^(1/2) =

• 1/2

• 1/3

• 1

• 2

Multiple Choice

log √1000 =

• 1/2

• 3/2

• 5/2

• 2

Multiple Choice

log 100√10

• 1/2

• 3/2

• 5/2

• 2

Multiple Choice

log 1000 + log 0.1 - log 0.01 =

• 0

• 2

• 3

• 4

Multiple Choice

log 100 + log 0.1 =

• 100.1

• 3

• 2

• 1

Logarithms to base a

• For  $a\ne1,\ a>0,\ y>0,$   $a^x=y\ \leftrightarrow\ x\ =\log_ay$  

• All rules in common logarithms also applied in this logarithms

Additional rules

•  $\log_{a^n}b^m\ =\frac{m}{n}\log_ab$  

•  $\log_ab\ =\frac{\log_cb}{\log_ca}$  

•  $\log_ab\ =\ \frac{1}{\log_ba}$  

•  $a^{\log_ab}\ =\ b$  

Examples

•  $\log_39=\log_33^2=2$  

•  $\log_93=\log_{3^2}3^1\ =\frac{1}{2}\log_33\ =\ \frac{1}{2}$  

•  $\log_{16}8\ =\log_{2^4}2^3\ =\frac{3}{4}\log_22\ =\frac{3}{4}$  

•  $2^{\log_25}\ =5$  

•  $2^{\log_83}\ =2^{\log_{2^3}3}\ =2^{\frac{1}{3}\log_23}\ =2^{\log_23^{\frac{1}{3}}}\ =\frac{1}{3}$  

Multiple Choice

Solve the equation for x.

• x = 27

• x = 9

• x = 3

• x = 1

Multiple Choice

Evaluate: log₉3 = x

• ½ = x

• -½ = x

• 2 = x

• -2 = x

Multiple Choice

Evaluate log₅25 = x

• x = 2

• x = 5

• x = 125

Multiple Choice

Write the following equation in logarithmic form:

3⁵ = 243

• log₃5 = 243

• log₂₄₃3 = 5

• log₅243 = 3

• log₃243 = 5

Multiple Choice

Rewrite the equation in exponential form:

log₅125 = 3

• 125³ = 5

• 5¹²⁵ = 3

• 5³ = 125

• 3⁵ = 125

The End

Thank you