It’s a Natural (Guided Practice)
The natural base, e, is an irrational number that approaches 2.71828…. An exponential function with base e is called a natural base exponential function, y = ex. Most calculators have an ex function for evaluating natural base expressions. Use a calculator to evaluate each expression to four decimal places. 1. e2 ≈ 7.3891
2. e−1.3 ≈ 0.2725
3. e5 ≈ 148.4132
4. e−2.2 ≈ 0.1108
The logarithm with base e is called the natural logarithm, sometimes denoted by loge x, but more often abbreviated ln x. The natural logarithmic function, y = ln x, is the inverse of the natural base exponential function, y = ex. Use a calculator to evaluate each expression to four decimal places. 1. ln 4 ≈ 1.3863
2. ln 0.05 ≈ −2.9957
3. ln 7 ≈ 1.9459
4. ln 0.25 ≈ −1.3863
Using the fact that ln x = loge x, write an equivalent exponential or logarithmic equation. (Remember: logb x = y if and only if by = x) 1. ex = 5 2. ln x ≈ 0.6931 3. ex = 6 4. ln x ≈ 0.5352
loge 5 = x ln 5 = x loge x ≈ 0.6931 x ≈ e0.6931 ln 6 = x x ≈ e0.5352
Since the natural base function and the natural logarithmic function are inverses, these two functions can be used to “undo� each other.
Equations and inequalities involving base e are easier to solve using natural logarithms than using common logarithms. All the properties of logarithms that you have learned apply to natural logarithms as well.
Solve 5e−x −7 = 2 5e−x = 9
Add 7 to each side.
e−x = 9/5
Divide each side by 5.
ln e−x = ln 9/5
Property of Exponents and Logarithms.
−x = ln 9/5 x = −ln 9/5
Inverse Property of Exponents and Logarithms. Divide each side by −1.
x ≈ −0.5878 Use a calculator. Now check your answer by substituting −0.5878 into the original equation or by finding the intersection of the graphs of y = 5e−x −7 and y = 2.
Solve 5e−x −7 = 2 5e−x = 9
Add 7 to each side.
e−x = 9/5
Divide each side by 5.
ln e−x = ln 9/5
Property of Exponents and Logarithms.
−x = ln 9/5 x = −ln 9/5
Inverse Property of Exponents and Logarithms. Divide each side by −1.
x ≈ −0.5878 Use a calculator. Now check your answer by substituting −0.5878 into the original equation or by finding the intersection of the graphs of y = 5e−x −7 and y = 2.
Solve 5e−x −7 = 2 5e−x = 9
Add 7 to each side.
e−x = 9/5
Divide each side by 5.
ln e−x = ln 9/5
Property of Exponents and Logarithms.
−x = ln 9/5 x = −ln 9/5
Inverse Property of Exponents and Logarithms. Divide each side by −1.
x ≈ −0.5878 Use a calculator. Now check your answer by substituting −0.5878 into the original equation or by finding the intersection of the graphs of y = 5e−x −7 and y = 2.
Solve 5e−x −7 = 2 5e−x = 9
Add 7 to each side.
e−x = 9/5
Divide each side by 5.
ln e−x = ln 9/5
Property of Exponents and Logarithms.
−x = ln 9/5 x = −ln 9/5
Inverse Property of Exponents and Logarithms. Divide each side by −1.
x ≈ −0.5878 Use a calculator. Now check your answer by substituting −0.5878 into the original equation or by finding the intersection of the graphs of y = 5e−x −7 and y = 2.
Solve 5e−x −7 = 2 5e−x = 9
Add 7 to each side.
e−x = 9/5
Divide each side by 5.
ln e−x = ln 9/5
Property of Exponents and Logarithms.
−x = ln 9/5 x = −ln 9/5
Inverse Property of Exponents and Logarithms. Divide each side by −1.
x ≈ −0.5878 Use a calculator. Now check your answer by substituting −0.5878 into the original equation or by finding the intersection of the graphs of y = 5e−x −7 and y = 2.
Solve 5e−x −7 = 2 5e−x = 9
Add 7 to each side.
e−x = 9/5
Divide each side by 5.
ln e−x = ln 9/5
Property of Exponents and Logarithms.
−x = ln 9/5 x = −ln 9/5
Inverse Property of Exponents and Logarithms. Divide each side by −1.
x ≈ −0.5878 Use a calculator. Now check your answer by substituting −0.5878 into the original equation or by finding the intersection of the graphs of y = 5e−x −7 and y = 2.
Solve 5e−x −7 = 2 5e−x = 9
Add 7 to each side.
e−x = 9/5
Divide each side by 5.
ln e−x = ln 9/5
Property of Exponents and Logarithms.
−x = ln 9/5 x = −ln 9/5
Inverse Property of Exponents and Logarithms. Divide each side by −1.
x ≈ −0.5878 Use a calculator. Now check your answer by substituting −0.5878 into the original equation or by finding the intersection of the graphs of y = 5e−x −7 and y = 2.
Solve each equation. 1. 3ex + 2 = 4
−0.4055
2. 4e−x − 9 = −2
−0.4055
When interest is compounded continuously, the amount A in an account after t years is found using the formula A = Pert, where P is the amount of principal and r is the annual interest rate. Suppose you deposit $1000 in an account paying 2.5% annual interest, compounded continuously. a) What is the balance after 10 years? A = Pert Continuous compounding formula = 1000e(0.25)(10) Replace P with 1000, r with 0.025, and t with 10. = 1000e0.25 Simplify. ≈ 1284.03 Use a calculator. The balance after 10 years would be $1284.03
When interest is compounded continuously, the amount A in an account after t years is found using the formula A = Pert, where P is the amount of principal and r is the annual interest rate. Suppose you deposit $1000 in an account paying 2.5% annual interest, compounded continuously. a) What is the balance after 10 years? A = Pert Continuous compounding formula = 1000e(0.25)(10) Replace P with 1000, r with 0.025, and t with 10. = 1000e0.25 Simplify. ≈ 1284.03 Use a calculator. The balance after 10 years would be $1284.03
When interest is compounded continuously, the amount A in an account after t years is found using the formula A = Pert, where P is the amount of principal and r is the annual interest rate. Suppose you deposit $1000 in an account paying 2.5% annual interest, compounded continuously. a) What is the balance after 10 years? A = Pert Continuous compounding formula = 1000e(0.25)(10) Replace P with 1000, r with 0.025, and t with 10. = 1000e0.25 Simplify. ≈ 1284.03 Use a calculator. The balance after 10 years would be $1284.03
When interest is compounded continuously, the amount A in an account after t years is found using the formula A = Pert, where P is the amount of principal and r is the annual interest rate. Suppose you deposit $1000 in an account paying 2.5% annual interest, compounded continuously. a) What is the balance after 10 years? A = Pert Continuous compounding formula = 1000e(0.25)(10) Replace P with 1000, r with 0.025, and t with 10. = 1000e0.25 Simplify. ≈ 1284.03 Use a calculator. The balance after 10 years would be $1284.03
b) How long will it take for the balance in your account to reach at least $1500? A = Pert
Continuous compounding formula
1500 ≥ 1000e0.025t
Replace A with 1500, P with 1000, r with 0.025
1500 1000 e0.025t ≥ 1000 1000
Divide each side by 1000.
1.5 ≥ e0.025t
Simplify
ln e0.025t ≥ ln 1.5
Property of Equality for Logarithms
b) How long will it take for the balance in your account to reach at least $1500? A = Pert
Continuous compounding formula
1500 ≥ 1000e0.025t
Replace A with 1500, P with 1000, r with 0.025
1500 1000 e0.025t ≥ 1000 1000
Divide each side by 1000.
1.5 ≥ e0.025t
Simplify
ln e0.025t ≥ ln 1.5
Property of Equality for Logarithms
b) How long will it take for the balance in your account to reach at least $1500? A = Pert
Continuous compounding formula
1500 ≥ 1000e0.025t
Replace A with 1500, P with 1000, r with 0.025
1500 1000 e0.025t ≥ 1000 1000
Divide each side by 1000.
1.5 ≥ e0.025t
Simplify
ln e0.025t ≥ ln 1.5
Property of Equality for Logarithms
b) How long will it take for the balance in your account to reach at least $1500? A = Pert
Continuous compounding formula
1500 ≥ 1000e0.025t
Replace A with 1500, P with 1000, r with 0.025
1500 1000 e0.025t ≥ 1000 1000
Divide each side by 1000.
1.5 ≥ e0.025t
Simplify
ln e0.025t ≥ ln 1.5
Property of Equality for Logarithms
b) How long will it take for the balance in your account to reach at least $1500? A = Pert
Continuous compounding formula
1500 ≥ 1000e0.025t
Replace A with 1500, P with 1000, r with 0.025
1500 1000 e0.025t ≥ 1000 1000
Divide each side by 1000.
1.5 ≥ e0.025t
Simplify
ln e0.025t ≥ ln 1.5
Property of Equality for Logarithms
0.025t ≥ ln 1.5
Inverse Property of Exponents and Logarithms
ln1.5 t≥ 0.025
Divide each side by 0.025.
t ≥ 16.22
Use a calculator.
It will take at least 16.22 years for the balance to reach $1500.
0.025t ≥ ln 1.5
Inverse Property of Exponents and Logarithms
ln1.5 t≥ 0.025
Divide each side by 0.025.
t ≥ 16.22
Use a calculator.
It will take at least 16.22 years for the balance to reach $1500.
0.025t ≥ ln 1.5
Inverse Property of Exponents and Logarithms
ln1.5 t≥ 0.025
Divide each side by 0.025.
t ≥ 16.22
Use a calculator.
It will take at least 16.22 years for the balance to reach $1500.
Suppose you deposit $5000 in an account paying 3% annual interest, compounded continuously. a) What is the balance after 5 years? $5809.17 b) How long will it take for the balance in your account to reach at least $7000? about 11.22 years
Solve each equation or inequality. a) ln 5x = 4 eln 5x = e4 5x = e4
e4 x= 5
Write each side using exponents and base e. Inverse Property of Exponents and Logarithms Divide each side by 5.
x ≈ 10.9196 Use a calculator. Check using substitution or graphing.
b) ln (x − 1) > −2 eln (x − 1) > e−2 x – 1 > e−2
Write each side using exponents and base e. Inverse Property of Exponents and Logarithms
x > e−2 + 1
Add 1 to each side.
x > 1.1353
Use a calculator. Check using substitution.
c) ln 3x = 7 â&#x2030;&#x2C6;365.5444 d) ln (3x + 2) < 5 {x| x < 48.8044)