Exponential Equations

One -to-One Proerties
a^x=a^y if and only if x=y
Logx=Logy if and only if x=y
Example
2^x=32
2^x=2^5
x=5

Inverse Properties
a^logx=x
loga^x=x
Example
e^x=7
lne^x=ln7
x=ln7

Strategies for Solving Exonential and Logarithmic Equations
1. Rewrite the given equation in a form to use the One-to-One properties of exponential or logarithmic functions.
2. Rewrite an exponential equation inlogarithmic form and applly the Inverse property of logarithmic functions.
3. Rewrite a logarithmic equation in exponential formand apply the Inverse Property of exponential functions.

Example
Solve 2(3^(2t-5))-4=11
2(3^(2t-5))-4=11 Write original equation
2(3^(2t-5))=15 Add 4 to each side
3^(2t-5)=15/2 Divide each side by 2
log'3 3^(2t-5)=log'3(15/2) Take log base 3 of each side
2t-5=log'3(15/2) Inverse Property
2t=5+log'3(7.5) Add 5 to each side
t=(5/2)+(1/2)log'3(7.5) Divide each side by 2
t~3.417 Use a calculator