Notes Exponential and Logarithmic Equations
Exponential Equations:
There are two types of exponential equations.
1. Exponential Equations where both sides can be expressed with the same base.
To solve these equations we use the one-to-one property. If:
ax  az
Then: x  z
a. Express both side of the equation with the same base
b. Set the “exponents” equal to each other and solve
Solve the following equations:
Ex1: 2 x  23
Ex2: 2 x1  23
x
2
x1
Ex3: 3
Ex5: 4 x 
Ex4: 8  4 x1
 81
1
2
2. Exponential Equations where both sides of the equation canNOT be expressed to the same base
To solve these equations:
a. Isolate the base (what you are raising to a power)
b. Take the ln or log of both sides of the equation (this causes the variable to pop
out of the exponent)
c. Solve the equation for x
Solve the following equations:
Ex6: 4  7
x
3 x 1
Ex8: 8
 5x
Ex7: 9 x  2
Ex8.2: 3e2 x 1  15
Ex8.3: 2x7  5  10
Logarithmic Equations:
There are two types of logarithmic equations.
1. Log equations where you use the definition of logs to solve log a x  y → a y  x
To solve these logarithmic equations you need to:
a. Condense down to a SINGLE logarithmic expression using properties of logs.
b. Use definition of logs to rewrite the equation as an exponential function
c. Solve for x
d. Check that the solution(s)doesn’t /don’t make the argument ≤ 0.
Solve the following logarithmic equations:
Ex9: log 4 x  3
Ex10: 3log8 x  1
2. Solving Log equations using the one-to-one property. If log a M  log a N then M  N
To solve these logarithmic equations you need to:
a. Condense down to a SINGLE logarithmic expression using properties of logs
on both sides of equal sign.
b. Set the arguments equal to each other and solve for x.
c. Check that the solution(s)doesn’t /don’t make the argument ≤ 0.
Solve the following logarithmic equations:
Ex11: log 2 5  log 2 ( x  3)  log 2 ( x  5)
Ex13: ln( x  1)  ln( x  9)  ln( x)
Ex12: ln( x  1)  ln( x)  ln(2 x)