Unit 7 - Exponentials and Logarithms

Miscellaneous

100

These are the points used to graph an exponential function quickly.

(1,b); (0,1); (-1,1/b)

200

These are the points used to graph a logarithmic function quickly.

(b,1); (1,0); (1/b,-1)

300

All logarithms have a base, but some are invisible. This base only appears when you ln.

e

400

It's easiest to change the question when asked to evaluate logarithmic expressions, but all we want is the value of log₄ 16.

2

500

In the formula A = P(1 + r/n)^nt, n represents this quantity.

The number of times the amount is compounded in a year.

Exponentials

100

This is the result of converting log 1000 = 3 to exponential form.

10³ = 1000

200

This is the result of converting log₉ 27 = x to exponential form.

9^x = 27

300

When graphing f(x) = 2^x-1 + 4, these are the points you would graph after transformations.

(2,6); (1,5); (0,9/2)

400

When graphing f(x) = 4^-x + 2, these are the points you would graph after transformations.

(-1,6); (0,3); (1,9/4)

500

When graphing f(x) = -(1/3)^x+3 - 2, these are the points you would graph after transformations.

(-4,-5); (-3,-3); (-2,-7/3)

Logarithms

100

This is the result of converting 6^x = 216 to logarithmic form.

log₆ 216 = x

200

This is the rule you can use to change 

log(xy) to log x + log y.

Multiplication Rule

300

This is the result of ln 1.

0

400

This is the result when expanding

log(3x⁶³⁴y²)

log 3 + 634 log x + 2 log y

500

This is the result of condensing

5 ln x - 3 ln y - 12 ln z

ln(x⁵ / (y³z¹²) )

Equations

100

This is the solution when solving the equation

log₃(x-2) = 3

x=29

200

This is the solution to the equation

5^2x = 23

x = (ln 23) / (2 ln 5)

300

This is the solution to the equation

log₄ x + log₄ (x - 3) = 1

x = 4

400

This is the answer when solving the equation

log(x+5) + log(x-4) = log(-4x-24)

no solution

500

This is the solution to the equation

5^{x^2-2x} = 125

x = 3,-1

Applications

100

Investing $2000 at a rate of 3.1% for 6 years compounded monthly gives this amount.

$2408.27

200

Investing $1500 at a rate of 5% compounded continuously yields this $2500 after this amount of time.

10.2 years

300

For a population of 110 million in 2023, the growth rate is 16% per year. The population will be 170 million in this year.

2025 or 2026

400

Half-life represents this.

The time it takes for a sample to decay to half of its starting amount.

500

Iodine 131 has a half-life of 8 days. Starting with 20 grams, this amount is left after 3 weeks.

3.24 grams (amounts within .1 gram are acceptable)