Download Logarithmic Functions

Section 3.4
1. Find each logarithm.
a. log5 25
b. log3 81
c. log3 ⅓
d. log3 1/9
e. log4 2
f. log2 ½
a. ln (e 10)
b. ln √e
c. ln 3√e
d. ln 1
e. ln (ln(ee))
f. lb (1/e 3 )
2. Find each logarithm.
3. Use the properties of natural logarithms to simplify the following function.
f(x) = ln (9x) – ln 9
4. Use the properties of natural logarithms to simplify the following function.
f(x) = ln (x 3) – ln x
5. Use the properties of natural logarithms to simplify the following function
f(x) = ln (x/4) + ln 4
6. Use the properties of natural logarithms to simplify the following function
f(x) = 8x – e lnx
7. Personal Finance: Interest - An investment grown at 24% compounded monthly. How many years will
it take to
a. double?
b. increase by 50%?
8. Personal Finance: Interest - A bank offers 7% compound continuously. How soon will a deposit:
a. triple?
b. increase by 25%
9. Personal Finance: Depreciation - An automobile depreciates by 30% per year. How soon will it be
worth only half its original value? [Hint: deprecation is like interest but at a negative rate.]
10. Business: Advertising - After t days of advertisements for a new laundry detergent, the proportion of
shoppers in a town who have seen the ads is 1 – e -0.03t. How long must the ads run to reach 90% of the
shoppers?
11. Behavioral Science: Forgetting - The proportion of students in a psychology experiment who could
remember an eight-digit number correctly for t minutes was 0.9 – 0.2 ln t (for t >1) Find the proportion
that remembered the number for 5 minutes.
12. General: Carbon 14 Dating - The proportion of carbon 14 still present in sample after t years is
e -0.00012t.
Estimate the age of the cave painting discovered which they were drawn contains only 2.3% of its
original carbon 14. They are the oldest known paintings in the world.
13. Social Science: education and income - According to a study, each additional year of education after
high school graduation increases one’s income by 16%. Therefore, with x extra years of education, your
income will be multiplied by a factor of 1.16 x.
How many additional years of education are required to double your income?
That is, find the x that satisfies 1.16 x = 2.
14. Solve the following exercise on a graphing calculator by graphing an appropriate
exponential function together with a constant function and using INTERSECT to find the solution.
General: Inflation - At 2% inflation, prices increase by 2% compounded continuously. How soon will
prices:
a. double?
b. triple?
15. General: Carbon 14 Dating - In 1991 two hikers in the Italian Alps found the frozen but well
preserved body of the most ancient human ever found, dubbed “Iceman”. Estimate the age of
Iceman if his grass cape contained 53% of its original carbon 14. (Use the carbon 14 decay
– 0.00012 t
function, f (t) = e
)