Download Section 16-1: Exponential Expressions, Equations, and Formulas

Section 16-1: Exponential Expressions, Equations, and Formulas
Learning Outcome 1
nt

r
Use the formula A = P  1 +  to find the compound amount (total paid back) for a loan of
 n
$10,000, for a term of 5 years at an annual interest rate of 8.4% with the interest compounded
semiannually.

r
A = P 1 + 
 n
nt
 0.084 
A = 10,000 1 +


2 
Substitute in the formula. P = $10,000; r = 0.084; n = 2; t =5
2( 5 )
A = 10,000(1 + 0.042)10
A = 10,000(1.042)10
A = 10,000(1.508958131)
A = $15,089.58
Simplify expression inside parentheses and exponent.
Further simplify expression inside parentheses.
Simplify power.
Perform multiplication.
The amount to be paid back is $15,089.58.
Learning Outcome 2
The formula A = Pe rt is the formula used to find the compound amount when the compounding is
continuous. A is the compound amount; P is the principal, r is the annual interest rate, and t is the
number of years of the loan or investment. If $10,000 is loaned with continuous compounding for 5
years at 8.4% annual interest, what is the compound amount?
A = Pen
Substitute appropriate values in formula.
0.084(5)
A = 10,000e
Simplify the exponent.
A = 10,000e0.42
Use a calculator to evaluate the formula.
A = $15,219.62
Learning Outcome 3
Solve the equation: 93x+2 = 272
93x+2 = 272
Write both sides of the equation with the same base.
2 3x + 2
3 2
(3 )
= (3 )
Multiply the exponents to simplify.
Perform the multiplication.
32(3x+2) = 33(2)
36x+4 = 36
If the bases are the same, the exponents will be equal. Write the
two exponents as an equation.
Solve the equation for x.
6x + 4 = 6
6x = 6 − 4
6x = 2
2
x=
6
1
x=
3
For x =
1
the original equation will be true.
3
Section 16-2: Logarithmic Expressions
Learning Outcome 1
Write 81 = 34 in logarithmic form. It is necessary to recognize the pattern that 3 is the base, 4 is
the exponent, and 81 is the power. Thus, in logarithmic form we have: log 3 81 = 4.
Learning Outcome 2
Write the expression log 15 225 = 2 in exponential form. We note that 15 is the base, 2 is the
exponent, and 225 is the power. Thus, we have 152 = 225.
Learning Outcome 3
Use a calculator to find log 42.
log 42 = 1.62324929
Learning Outcome 4.
Use your calculator to evaluate log 7 343.
log a
log 343
.
log 7 343 =
=3
Use calculator and log function by applying the rule logb a =
log 7
log b
Verify the result by raising 7 to the 3rd power.
7(7)(7) = 343.
Learning Outcome 5
 I
to find the decibel rating for a sound that is 200 times the
 I0 
Use the formula bel = 10 log
threshold sound (200I0).
 I

 I0 
 200I0 
bel = 10 log

 I0 
bel = 10 log
bel = 10 log (200)
bel = 23.01
Write the formula, and substitute for I.
I0 cancels.
Find log 200 and multiply by 10.
Learning Outcome 6
Use the laws of exponents to evaluate the following logarithm expressions.
log (5)(12)
log (5)(12) = log 5 + log 12
Rewrite the indicated multiplication as addition.
log (5)(12) = 0.6989700043 + 1.079181246 = 1.77815125
Rewrite as the sum of two logarithms: ln (3)(5)
ln (3)(5) = ln (3) + ln (5)
ln (3)(5) = 1.098612289 + 1.609437912 = 2.708050201