Logarithms

Converting Between Forms

100

2³=8

What is the logarithmic form?

log₂8=3

200

log₅25=2

What is the exponential form? 

5²=25

300

log₄x=3

What is x? 

x=64

400

10^x=1,000,000

What is log₁₀1,000,000

6

500

log₃81=x

What is the exponential form?

3^x=81

Evaluating Logarithms

100

log₁₀100 = ?

2

200

log₂16 = ? 

4

300

log₃1/27 = ?

-3

400

log₅sqrt(5) = ?

1/2

500

log₇49 = ?

2

Condense & Expand

100

Expand: log(4x)

log4 + logx

200

Expand: log(3x²) 

log3+2logx 

300

Condense: logx + logy 

log(xy)

400

Condense: 2loga - logb

log(a²/b)

500

Expand: log((x³y)/z)

3logx + logy - logz

Logarithmic & Exponential Equations

100

Solve: logx = 3

x=1,000

200

Solve: 2^x = 32 

x=5

300

Solve: log₂(x) = 5 

x=32

400

Solve: e^x = 20

x=ln20

500

Solve: ln(x) = 2

x=e²

Real-World Applications

100

In the Ritchter scale, an earthquake that measures 6 is how many times stronger than one that measures 5?

10 times

200

A sound at 60 dB is how many times louder than one at 30 dB? 

(Each 10 dB = x10)

1000 times louder

300

If a $500 investment grows exponentially ro $2000 in 10 years, find rate r using 2000 = 500e^10r

r = 0.1386 or 13.86%

400

A radioactive substance decays according to               A = A₀e^-0.3t

When will half remain?

t = 2.31 years

500

The pH of a solution is 4. 

What is its hydrogen ion concentration?

(pH = -log[H⁺])

[H⁺] = 10^-4M