Logarithms

Exponential & Logarithmic Forms

100

Write the exponential equation as a logarithmic equation. 2⁸ = 256

What is log₂(256) = 8?

200

Write the logarithmic equation as an exponential equation. log₉(3) = ½

What is 9^½ = 3?

300

Evaluate log₂ 8.

What is 3?

400

Solve for x: log⁡₅(2x+3)=4

What is 311?

500

Evaluate. log₄32

What is ⁵/₂?

Logarithmic Properties & Simplification

100

log₃2+log₃5.

What is log₃10

200

2 log x - log y

What is? log (x²/y)

300

Condense: log₁₀x+3log₁₀y−log₁₀z

log⁡₁₀(xy³/z)

400

Compute (and simplify): log₂8+1/2log₂16-log₂4

What is 3?

500

Solve for x by condensing: log₅x+log₅(x-4)-2log₅2=0

What is x=2+2√2?

Logarithmic Equations

100

log₁₀x=2

What is 100?

200

log₂(x-1)=3

What is 9?

300

log x + log(x-3)=1

What is 5?

400

ln(2x+1)=3

What is (e³-1)/2?

500

log₃(x²-x)=4

What is (1±√325)/2?

Exponential Equations & e/ln

100

2^x=16

What is 4?

200

5e^2x=20

x=ln4/₂

300

3^2x-1=27

What is 2?

400

1000=50e^0.03t

t=ln20/_0.03

500

7^x+1=5^2x

What is ln7/2ln5−ln7?

Real World Applications

100

A solution has [H⁺] = 1 x 10^-6 M? What is the pH level?

(The formula being pH = -log[H⁺].)

What is 6?

(fun fact we actually did this in chemistry class!)

200

A culture starts with 200 bacteria and grows exponentially at 8% per hour. Write the exponential model.

What is N(t)=200(1.08)^t?

300

You invest $1,000 at 5% annual interest, compounded annually. How much will you have after 10 years?

What is $1628.89?

400

A radioactive substance decays exponentially and has a decay constant k=0.12. If the initial mass is 50 g, how much remains after 5 hours?

What is 27.44 g?

500

A population follows P(t)=P₀e^0.03t It takes 12 years to reach 2,000 organisms. Find the initial population P₀.

What is 1393.5?