5.4 Exponential and Logarithmic Equations

A. One-to-one and inverse properties

B. Solve exponential equations

C. Solve log equations

A. One-to-one and inverse properties

• The one-to-one property in this section is just a little different from before.

• logax = logay would imply that x = y.

• lnx – ln(x + 2) = ln(x + 6)

Inverse Property:

• Remember that neato product property of logs? log 3x =

• What power! We can bring exponents down! So what if we had an exponential equation like: 3x = 14

• We cannot use the one-to-one property on it because we cannot write 14 as an expression with base 3. If only….

3x = 14

• If only there were a “log” or an “ln” there…. Then we could bring down the x….

• WE CAN TAKE THE “LOG” OR “LN” OF BOTH SIDES!!

• ln 3x = ln 14 Then product property.• x ln3 = ln 14 Then solve for x…

[Remember that things like “ln3” and “ln14” are just numbers.]

B. Exponential Equations when you can’t solve using one-to-one

• STEP 1: Get the exponential expression by itself (if it isn’t already)

• STEP 2: Take the log or ln of both sides.• STEP 3: Pull the exponent down.

• STEP 4: Solve for x.

( ) 4038 6 =−x

2140

471.24

9

=

−

t

( ) 7105: 6 =−xTry

C. Solve log Equations

• We’ve done each of these before. • IF there are more than one logarithmic

expression on one side of the equal sign, then try condensing them to make it one.

• IF there are two logs, one on each side of the equals, then try using the one-to-one property (objective A of this lesson).

• IF there is only one log on only one side of the =, then convert it to exponential form.

( ) ( )6ln2lnln

1ln4ln

+=++

=+

xxx

x