6.4-6.5 Review

Expanding Logs

100

log₂(x³/y)

3log₂x - log₂y

200

log₃(a/b²c³)

log₃a - 2log₃b - 3log₃c

300

log(x²yz⁴)^1/2

log x + (1/2)log y + 2log z

400

log₅(x∛y²/z⁴)

log₅x + (2/3)log₅y - 4log₅z

500

log₇(x²/4y)^-3

log₇64 + 3log₇y - 6log₇x

Condensing Logs

100

log₃x + log₃y + log₃z

log₃(xyz)

200

(1/2)log x - 3log y⁴

log (√x / y¹²)

300

2log₄(x - 3) + 2log₄x

log₄(x⁴ - 6x³ + 9x²)

400

(1/3)log₂x - log₂y + (1/4)log₂z³

log₂(∛x∜z³/y)

500

(1/3)log x + (1/3)log₃y + (2/3)log z

log∛xz² + log₃∛y

Evaluating Exponentials

100

3^x = 9⁴

x = 8

200

4^x = 17

x = 2.044

300

16 = 1 + 3⋅5^2x

x = 1/2

400

(1/9)^-r = 27^3r-2

r = 6/7 or 0.857

500

2 ⋅ 4³ = 8^x

x = 7/3

Evaluating Logarithms

100

5^log₅^(2x)

2x

200

log₇x + 3 = 2

x = 1/7

300

2log₃x = log₃(5x + 6)

x = 6

400

log₃(4x) = log₃(2x) + log₃(x+1)

x = 1

500

1 + log₄x = 3log₄(2x)

x = √2/2

Other

100

Find the inverse of y = 4^x.

f ^-1(x) = log₄x

200

The population P of a city is modeled by 𝑃(𝑡) = 10,000𝑒^rt. If the population increased to 30,000 over 5 years, determine the percent of growth.

r = 22%

300

Find the inverse of f(x) = e^3x - 1

f ^-1(x) = ln(x+1)/3

400

Andy wants to invest $3,500 in a bank. If he wants to double the amount in 8 years, what must the annual rate be if the interest is compounded continuously?

r = 87% or 0.087

500

The population of a type of bacteria doubles every 3 hours, which can be modeled by 𝑛(𝑡) = a ∙ 2^rt, where t represents time in hours. Find the rate.

r = 33% or 0.33