Crash Course - Logarithms

​Crash Course;
Logarithm Basics

By William Katumba

Why logarithms?

Do we use logarithms in real life?

Multiple Choice

$$2^5=32$$  Which part is the base?

• 2

• 5

• 32

Multiple Choice

$$2^5=32$$  Which part is the exponent?

• 2

• 5

• 32

Multiple Choice

$$2^5=32$$  Which part is the solution (answer)?

• 2

• 5

• 32

Multiple Choice

$$2^5=32$$  What is the logarithmic form?  $$\log_{base}\left(answer\right)=\exp onent$$  

• $$\log_52=32$$  

• $$\log_{32}5=2$$  

• $$\log_232=5$$  
  

Multiple Select

$$8^2=64$$  Can be written in logarithmic form as...

• $$\log_28=64$$  

• $$\log_864=2$$  

• $$\log_82=64$$  

Fill in the Blanks

$4^3=64$  

Write the exponential equation in logarithmic form.

$\log_4\left(\ \ ?\ \ \right)=3$  

Fill in the Blanks

$7^3=343$  

Write the exponential equation in logarithmic form.

$\log_7\left(\ \ \ \ ?\ \ \ \ \ \right)=3$  

Fill in the Blanks

$10^3=1000$  

Write the exponential equation in logarithmic form.

$\log_{10}1000=\left(\ \ \ \ ?\ \ \ \ \right)$  

Fill in the Blanks

$5^4=625$  

Write the exponential equation in logarithmic form.

$\log_5625=\left(\ \ \ ?\ \ \ \right)$  

Multiple Choice

Solve for x
$$\log_2x$$   = 1/2

• $$\sqrt{2}$$  

• -2

• -1

• 1

Multiple Choice

Solve for x
$$\log_2\left(\frac{1}{16}\right)$$   = x

• 4

• -4

• 8

• -8

Multiple Choice

The common logarithm has what base?

• 10
• 0
• e
• -e

Multiple Choice

If 3^x = y, which of the following is true?

• log₃y = x
• log₃x = y
• log_xy = 3
• log_y3 = x

Properties of Logarithms

Multiple Choice

Condense

$$\log_35+\log_34$$  

• $$\log_39$$  

• $$\log_320$$  

• $$\log_69$$  

• $$\log_620$$  

Multiple Choice

Condense

$$\log_53+\log_5y+\log_5z$$  

• $$\log_53yz$$  

• $$\log_{15}3yz$$  

• $$\log_35yz$$  

• $$\log_z15y$$  

Multiple Choice

Expand

$$\log_28y$$  

• $$\log_82+\log_8y$$  

• $$\log_28+\log_2y$$  

• $$\log_2y+\log_82$$  

• $$\log_y2+\log_y8$$  

Multiple Choice

Solve
$$\log_73+\log_7y=\log_718$$  

• 6

• 54

• 1/6

• 21

Multiple Choice

Solve
$$\log_{10}\left(3m-5\right)+\log_{10}m=\log_{10}2$$  

• -5

• 7/4

• 2, -1/3

• -2, 1/3

Quotient Property

Multiple Choice

Condense

$$\log_7y-\log_79$$  

• $$\log_7\frac{9}{y}$$  

• $$\log_79y$$  

• $$\log_7\frac{y}{9}$$  

• $$\log_9\frac{y}{7}$$  

Multiple Choice

Expand

$$\log_4\frac{y}{10}$$  

• $$\log_4y+\log_410$$  

• $$\log_410-\log_4y$$  

• $$\log_{10}y-\log_{10}4$$  

• $$\log_4y-\log_410$$  

Multiple Choice

Solve

$$\log_848-\log_8y=\log_84$$  

• 1/12

• 12

• 44

• 192

Multiple Choice

Solve

$$\log_8\left(t+10\right)-\log_8\left(t-1\right)=\log_812$$  

• 1

• 2

• 11

• 22

Power Property

Multiple Choice

Condense

$$3\log_47$$  

• $$\log_47^3$$  

• $$\log_421$$  

• $$\log_43^7$$  

• $$\log_410$$  

Multiple Choice

Expand

$$\log_2^{ }y^5$$  

• $$y\cdot\log_25$$  

• $$5\log_2y$$  

• $$2\cdot\log_5y$$  

• $$\log_25^y$$  

Multiple Choice

Solve

$$3\log_74=2\log_7b$$  

• 64

• 4

• 8

• 16

Multiple Choice

Solve

$$3\log_82-\log_84=\log_8g$$  

• 2

• 4

• 8

• 12

Multiple Choice

Solve

$$\log_2w+2\log_25=0$$  

• 0

• 10

• 25

• 32

Multiple Choice

Solve

$$\log_2w+2\log_25=0$$  

• 0

• 10

• 25

• 32

Multiple Choice

Solve

$$\log_3y=-\log_316+\frac{1}{3}\log_364$$  

• 4

• 16

• 1/4

• 1/2

Multiple Choice

Evaluate the logarithm. Round your answer to the nearest thousandth. (3 decimal places)

$$\log_8200$$  

• $$\log_8200=\frac{\log200}{\log8}\approx2.548$$  

• $$\log_8200=\frac{\log8}{\log200}\approx.392$$  

Changing the base

Fill in the Blanks

Evaluate the logarithm. Round your answer to the nearest thousandth. (3 decimal places)

$\log_7\left(\frac{3}{5}\right)$  

Multiple Select

Use the Change of Base Property to evaluate the following logarithms. Then use Desmos to verify.

$$\log_29=$$

• $$\frac{\log2}{\log9}$$

• $$\frac{\log9}{\log2}$$

• 0.3154

• 3.1699

Multiple Select

Use the Change of Base Property to evaluate the following logarithms. Then use Desmos to verify.

$$\log_{1000}100=$$

• $$\frac{\log100}{\log1000}$$

• $$\frac{\log1000}{\log100}$$

• 1.5

• 0.6667

Multiple Select

Use the Change of Base Property to evaluate the following logarithms. Then use Desmos to verify.

$$\log_5200=$$

• $$\frac{\log5}{\log200}$$

• $$\frac{\log200}{\log5}$$

• 3.292

• 0.3037

Multiple Choice

Use the change of base rule to rewrite this problem:

log₇729

• log(7)/log(729)

• log(729)/log(7)

Multiple Choice

Use the change of base rule to rewrite this problem:

log₅75

• log(75)/log(5)

• log(5)/log(75)

Multiple Choice

Use the change of base rule to rewrite this problem:

log₃64

• log(64)/log(3)

• log(3)/log(64)

Multiple Choice

Solve the log using change of base
log₉27

• 3/2
• 2/3
• 3
• 0.5

Multiple Choice

Solve the log by using change of base )

log₄32

• 8

• 5/2

• 2/5

• 3/2

Multiple Choice

Using the change of base rule, what is this logarithm equivalent to?

• A
• B
• C
• D

Multiple Choice

Use the change of base rule to rewrite this problem:

log₄92

• $$\frac{\log\left(4\right)}{\log\left(92\right)}$$

• $$\frac{\log\left(92\right)}{\log\left(4\right)}$$

• $$\frac{\log\left(92\right)}{\ln\left(4\right)}$$

• $$\frac{\log\left(4\right)}{\ln\left(92\right)}$$

Multiple Choice

Use the change of base rule to rewrite this problem:

log₇256

• $$\frac{\log_5\left(256\right)}{\log_3\left(7\right)}$$

• $$\frac{\log_2\left(256\right)}{\log_2\left(7\right)}$$

• $$\frac{\ln\left(7\right)}{\ln\left(256\right)}$$

• $$\frac{\ln\left(256\right)}{\log\left(7\right)}$$

Multiple Select

Use the definition of the natural logarithm function, as well as properties of logarithms, to simplify or expand each expression.

$$\ln\left(\left(2e\right)^3\right)=$$

Use properties of logarithms

• $$\ln2^3+\ln e^3$$

• $$3\ln2+3\ln e$$

• $$3\ln2+\ln3e$$

• $$6\ln2e$$

Multiple Select

Use the definition of the natural logarithm function, as well as properties of logarithms, to simplify or expand each expression.

$$\ln\left(2e^3\right)=$$

Use properties of logarithms

• $$\ln5e$$

• $$\ln2+3\ln e$$

• $$\ln2+\ln e^3$$

• $$\ln6e$$

Multiple Select

Use the definition of the natural logarithm function, as well as properties of logarithms, to simplify or expand each expression.

$$\ln e^2=$$

Use properties of logarithms

• 1

• $$2\ln e$$

• $$\ln e$$

• 2

• $$\ln e\ln e$$

Multiple Choice

Use the definition of the natural logarithm function, as well as properties of logarithms, to simplify or expand each expression.

$$\ln e=$$

Hint: type into Desmos if you are not sure

• e

• 0

• 1

Fill in the Blanks

What is the shorthand notation for $y=\log e$ ?

$y=$

Match

Match the following by solving for x.

• $$\log_x27=3$$

• $$4^x=2$$

• $$\log_3x=-2$$

• $$8^{\frac{2}{3}}=x$$

• $$\log_8x=1$$

3

1/2

1/9

4

8

• 3

• 1/2

• 1/9

• 4

• 8

Multiple Choice

Evaluate:

$$\ln e^2=x$$  

• $$1$$  

• $$2$$  

• $$e$$  

• $$4$$  

Multiple Choice

Solve the equation.

$$\ln\left(3x\right)=2$$  

• $$x=2.5$$  

• $$x=7.4$$  

• $$x=2.7182818...$$  

• No Solution