When you have completed this section you should be able to:

- Solve exponential equations analytically
- Solve logarithmic equations analytically
- Identify exponential/logarithmic equaitons that cannot be solved analytically and solve them graphically or numerically

Simple Equations

- Exponential
Given: 35 = 2

^{3x}- First take the logs (or the natural log - ln) of both sides of the equation.
log(35) = log(2

^{3x})- Now move the exponent outside the log on the right hand side using the fact that log( a
^{p}) = p log(a)log(35) = 3x ( log(2) )

- Divide both sides of the equation by 3(log(2))
x = log(35)/(3 (log(2))

- Calculate x using your calculator. x = 1.7098
Now you try it using Practice Problem 1: 68 = 4

^{5x}(answers at the end of this section)

- Logarithm
Given: log(x+3) = 2

- Using the relationships log(M) = N and M = 10
^{N}to convert the log equation to the power equation x+3 = 10^{2}- Simplify:

- x + 3 = 100
- x = 100 - 3
- x = 97

Now you try it using Practice Problem 2: log(x^{2} - 8) = 0
(answers at the end of this section)

More Complicated Equations

ExponentialGiven: 3

^{x}= 4^{(x+1)}- First note that both exponents involve the variable (this is good news) Take the log (or ln ) of both sides of the equation.
log(3

^{x}) = log( 4^{(x+1)})- Now using the property of logs that says log( a
^{p}) = p log(a) pull the exponents out of the log( )x log(3) = (x+1) log (4)

- Perform the multiplication called for on the right side of the equation.
x log(3) = x log(4) + 1 log(4)

- Move x log(4) to the left hand side of the equation.
x log(3) - x log(4) = log(4)

- Factor the x (it is a common term on the left side of the equation)
x [log(3) - log(4)] = log(4)

- Divide both sides of the equation by [log(3) - log(4)]
x = log(4)/[log(3) - log(4)]

- Now calculate the result, x = -4.8188
And here is Practice Problem 3: 5.1

^{(x+2)}= 8.7^{(x -1) }

## Practice Problem Answers:

Practice Problem 1: 68 = 4

^{5x}- log( 68 ) = log( 4
^{5x})- log( 68 ) = 5x log( 4 )
- x = log( 68 )/(5 log(4))
- x = 0.60875

Practice Problem 2: log(x

^{2}- 8) = 0- x
^{2}- 8 = 10^{0}- x
^{2}- 8 = 1- x
^{2}= 1 + 8 = 9- x = + 3 AND X = -3 (do both roots work in the original equation?)

Practice Problem 3: 5.1

^{x+2}= 8.7^{(x -1)}- log (5.1
^{x+2}) = log (8.7^{(x -1)})- (x+2) log(5.1) = (x - 1) log(8.7)
- x log(5.1) + 2 log (5.1) = x log(8.7) - log(8.7); Move "like" terms to the same side of the equation.
- x log(5.1) - x log(8.7) = -2log(5.1) - log(8.7), Now factor the x from the left hand side of the equation.
- x [log(5.1) - log(8.7)] = -2log(5.1) - log(8.7), divide by the coefficient of x which is [log(5.1) - log(8.7)]
- x = [-2log(5.1)-log(8.7)]/[log(5.1) - log(8.7)]
- x = 10.1516