1. Chapter 5
Logarithmic Functions

2. 5.4 Properties of Logarithms

3. Exponential/Logarithmic Forms Property
For a > 0, b > 0, and b ≠ 1, the equations
logb(a) = c and bc = a
are equivalent.

4. Example: Solving Logarithmic Equations in One Variable
Solve for x.
1. 6 log9(t) + 1 = log3(x4) = 2

5. Solution
1. Get log9(t) alone on the left side of the equation and solve for t:

6. Solution
2. Write log3(x4) = 2 in exponential form and solve for x:

7. Example: Solving for the Base of a Logarithm
Solve for b.
1. logb(81) = logb(67) = 5

8. Solution

9. Summary
For an equation of the form logb(x) = k, we can solve for b or x by writing the equation in exponential form.

10. Power Property for Logarithms
For x > 0, b > 0, and b ≠ 1,
logb(xp) = p logb(x)
In words, a logarithm of a power of x is the exponent times the logarithm of x.

11. Logarithm Property of Equality
For positive real numbers a, b, and c, where b ≠ 1, the equations
a = c and logb(a) = logb(c)
are equivalent.

12. Example: Solving an Exponential Equation
Solve the equation 2x = 12.

13. Solution
Check: ≈ ≈ 12

14. Computing a Quotient with a Calculator
Warning
To compute the quotient in the previous example:

15. Example: Solving an Exponential Equation
Solve 3(4)x = 71.

16. Solution
Check:
3(4) ≈
≈ 71

17. Dividing Both Sides of an Equation
Warning
To solve some equations of the form abx = c for x, we divide both sides of the equation by a, and then take the log of both sides. Next, use the power property for logarithms.
Note that in the previous example,
3(4)x ≠ (3∙4)x

18. Example: Solving an Exponential Equation
Solve 7(2)x – 4 = (2)x.

19. Solution

20. Solution
To check on a graphing calculator, graph the equations y = 7(2)x – 4 and y = (2)x and find the approximate intersection point (2.5850, 38), which has x-coordinate This checks.