Solving Exponential and Logarithmic Equations

One-to-One Properties:

if and only if x=y

if and only if x=y

Inverse Properties:

=x


=x

Solving Exponential and Logarithmic Equations

- rewrite the equation using one to one properties

Ex.

x=3

- rewrite an exponential equation as a logarithmic equation and use the inverse properties

Ex.

since there is an e in the equation, we use natural log (ln) rather than log

the ln cancels out the e since an e is already included when we use ln

x= ln5

-Rewrite a logarithmic equation as an exponential equation and use the inverse properties

log x= 2

10^(log x) = 10^2

the 10 and the log cancel out

x= 10^2

x=100

Examples from class:

Ex. 1

2+3^(x-4) = 12

subtract the 2

3^(x-4) = 10

change to a logarithmic equation ( the exponent (x-4) will be multiplied by the logarithm)

(x-4) log 3= log 10

log 10 is equal to one (10^1 = 10)

(x-4) log 3= 1

divide by the logarithm

(x-4) = 1/(log 3)

add the 4

x= 1/(log 3) + 4

solve using a calculator

x= 6.10

Ex. 2

e^(2x) - 3e^x +2 = 0

let u = e^x

u^2 - 3u + 2 = 0

factor the equation

(u-1)(u-2) = 0

u=1 u=2

e^x=1 e^x=2

x=0 ln(e^x) =ln 2

x=.7