Logarithms

Converting between exponential and logarithic form

100

Convert this logarithic into exponential form

log₂¹²⁸⁼⁷

What is.......

2⁷=128

200

Convert this logarithic into exponential form

log₈64=2

What is.......

8²=64

300

Convert from exponential form into log form

8³=512

What is.......

log₈512=3

400

Convert the log equation to an exponential equation

In2x=1/8

What is....

e^1/8=2x

500

Convert the exponential equations into a log equation

e⁹=x+2

What is.....

In(x+2)=9

Evaluating logarithms

100

Evaluate the logarithm (solve)

log₆36=

What is......

6^x=36

6²=36

x=2

200

Evaluate the logarithm (solve)

log₂32=

What is......

2^x=32

2⁵=32

x=5

300

Evaluate the logarithm (solve)

log₄64=

What is........

4^x=64

4³=64

x=3

400

Evaluate the logarithm (solve)

log₃81=

What is.....

3^x=81

3⁴=81

x=4

500

Evaluate the logarithm (solve)

log₁₀₀10=

What is.....

100^x=10

100^1/2=10

x=10

Condensing logarithms

100

Condense into a single logarithm

log₂7+log₂4

What is.....

log₂28

200

Condense into a single logarithm

log₃15+log₃4x

What is....

log₃60x

300

Condense into a single logarithm

log(x²) - log(y³)

What is....

log₁₀(x²/y³)

400

Condense into a single logarithm

log₂25+log₂(x²+10x)

What is....

log₂(25x²+250x)

500

Condense into a single logarithm

4In(x+6)-3Inx

What is.....

In((x+6)⁴/x³)

Random Log equations

100

Solve the logarithmic equation

log₃(x+5)=4

What is....

3⁴=x+5

200

Solve the logarithmic equation

In1

What is.....

ln is the same as log_e

log_e(e⁰)

0

300

Solve the logarithmic equation

log1/100

What is....

log₁₀(1/10²)

log₁₀(10^-2)

-2

400

Solve the logarithmic equation

log₄(4^1/5)

What is....

1/5

500

Solve the logarithmic equation

log₃₂(1/64)

What is......

x=-6/5

real-world applications

100

 The formula is B= 10Log(I/I₀) where B is the sound of decibels, I is the sound intensity, and I₀ is the reference intensity (threshold of human hearing). A whisper has a sound intensity of 10-10W/m2. How many decibels is it?

What is....

B=10log(10^-10/10^-12)

B=10log(10²)

B=10(2)=

20dB

200

The formula is M=log₁₀(A/A₀) where M is the magnitude, A is the amplitude of the seismic waves, and A₀ is a reference amplitude. An earthquale has a seismic waves amplitude 100 times greater than the reference amplitude. What is the magnitude on the Richter scale.

What is.....

M=log10(100xA₀/A₀)

M=log10(100)

M=2

300

Formula is pH=-log[H⁺], where [H⁺] is the hydrogen ion concentration in moles per liter. Lemon juice has a hydrogen ion concentration of 0.001 M. What is its pH level?

What is....

pH=-log(0.001)

pH=-log(10^-3)

pH=-(-3)=3

400

Formula is P(t)=P₀(1+r)^t, where P(t) is the population at time t, P₀ is the initial population, and r is the annual growth rate. To solve for t, you use logarithms. A city with an initial population of 50,000 grows at a rate of 3% per year. How long will it take for the population to double to 100,000?

100,000=50,000(1.03)^t

2=(1.03)^t

log(2)=t log(1.03)

t~0.06931/0.02956

t=~23.449 years

500

Formula: m₁-m₂= 2.5 log (b₂/b₁). This shows the relationship between the magnitude (m) and the apparent brightness (b) of two stars. If Sirius has a magnitude of -1.5 and Rigel has a magnitude of 0.12, how many times brighter is Sirus than Rigel?

What is.....

m_sirius-m_rigel=-1.5-0.12=-1.62

b_sirius/b_rigel=10^0.2(-1.62)=10^-0.324

b_siruis/b_{rigel~~ 0.47}

_{Sirius is about 0.47 times as bright as Rigel. Sirius is more than twice as bright as Rigel.}