Download 1014 Sec. 4.4 Notes

Math 1014: Precalculus with Transcendentals
Ch. 4: Exponential and Logarithmic Functions
Sec. 4.4: Exponential and Logarithmic Equations
I.
Exponential Equations
A. Definition
An exponential equation is an equation containing a variable in an exponent.
B. Solving Exponential Equations by Expressing Each Side as a Power of the Same Base
If b M = b N , then M = N.
1. Rewrite the equation in the form
2. Set
M=N.
bM = bN .
3. Solve for the variable.
C. Examples
Solve for x:
1.
3x = 81
2.
2 4 x−1 = 8
D. Using Logarithms to Solve Exponential Equations
1. Isolate the exponential expression.
10. Take the
natural logarithm on both sides of the equation for bases other than 10.
2. Take the common logarithm on both sides of the equation for base
3. Simplify using one of the following properties:
ln(b x ) = x ln(b) or ln(e x ) = x or log (10 x ) = x
4. Solve for the variable.
E. Examples
Solve for x:
1.
10 x = 8.07
2.
9e x = 107
II.
3.
5 x−3 = 137
4.
e4 x − 3e2 x − 18 = 0
Logarithmic Equations
A. Definition
A logarithmic equation is an equation containing a variable in a logarithmic expression.
B. Using the Definition of a Logarithm to Solve Logarithmic Equations
1. Express the equation in the form
log b M = c .
2. Use the definition of a logarithm to rewrite the equation in exponential form:
log b M = c ⇔ b c = M .
3. Solve for the variable.
4. Check proposed solutions in the original equation. Include in the solution set only
values for which
C. Examples
Solve for x.
1.
log 5 x = 3
M >0.
2.
log 6 (x + 5) + log 6 x = 2
3.
log 2 (x − 3) + log 2 x − log 2 (x + 2) = 2
Solve, leaving your answer in terms of e.
4.
ln x = 3
5.
6(ln(2x) = 30