Logarithms

Logarithmic Functions

100

Change the exponential equation to a log. 

6⁴=1296

log₆1296=4

200

Evaluate log₃64. Round to the nearest ten-thousandth.

3.7856

300

Change the logarithm to exponential.

log₉729=3

9³=729

400

Solve log₁₁(x²+5x-108) = log₁₁28x

Hint: factor a quadratic and check solutions

27

500

State the domain, range, and asymptote

f(x)=log₈(x+2)+1

Domain: x>-2

Range: all real numbers

Asymptote: -2

Properties of Logs

100

What two operations go along with the product property of logarithms?

Multiplication

Addition

200

What two operations go along with the quotient property of logarithms.

Division

Subtraction

300

Expand the logarithm:

log₃5²

2log₃5

400

Condense the logarithm into a single log:

2log(3) + log(x)

log(9x)

500

Condence the log into a single log:

log₂5 + 3log₂2 - log₂15

log₂40/15

Common Logs

100

Commons logs have what number for a base?

10

200

Evaluate log0.005. Round to the nearest ten-thousandth.

-2.3010

300

Write the log as a common log. 

log₅17

log17/log5

400

Solve 9^x=41. Round to the nearest ten-thousandth.

1.6901

500

Solve 14^x-5=8^x. Round to the nearest ten-thousandth

23.5792

Natural Logs

100

Natural logs (ln) are the opposite of what base in an exponential function?

e

200

Write the exponential equation as a logarithm. 

e^9.7=12x

ln12x=9.7

300

Solve for x. Round to the nearest ten-thousandth.

e^5x=37

0.7222

400

Solve for x. Round to the nearest ten-thousandth.

3e^x-7=19

2.1595

500

Solve for x. Round to the nearest ten-thousandth.

ln(x+7)=4

47.5982

Using Logs

100

What is the difference in equations between continuous growth and continuous decay?

Decay has a negative rate (k value), growth does not

200

What is the half-life equation when no percent is given?

What is the half-life equation when percent is given?

0.5a=ae^-kt

%a=ae^-kt

300

The half-life of Colbalt-60 is 5.27 years. Write the equation of decay for Colbalt-60. Round your decimals to the nearest ten-thousandth.

y=ae^-0.1315t

400

In 2010, the population of a town was 38,655 people. The populaton steadily increased to 41,204 in 2020. Write an equation to represent the population of the town after t years. Round the k value to five decimal places. 

y=38655e^0.00639t

500

Carbon-14 has a decay rate of 0.00012. A fossilized tree contained Carbon-14 has lost 35% of it's Carbon-14. How old is this tree? Round to the nearest year.

3590 years