exponential and logarithm equations in this

Question

Exponential and Logarithm Equations : In this section we'll learn solving equations along with exponential functions or logarithms in them. We'll begin with equations which involve exponential functions. The main property that we'll require for these equations is,

                                                                log_b bx  = x

Example   Solve following 7 + 15e^{1-3 z} = 10 .

Solution : The primary step is to get the exponential all by itself on one side of the equation  along with a coefficient of one.

7 + 15e^{1-3 z}  = 10

15e^{1-3 z}  = 3

e^{1-3 z}  = 1/5

Now, we have to get the z out of the exponent therefore we can solve for it. To do this we will employ the property above.  As we have an e in the equation we'll employ the natural logarithm.  Firstly we take the logarithm of both sides and then employ the property to simplify the equation.

ln (e^{1-3 z} ) = ln ( 1/5 )

1 - 3z = ln ( 1/5 )

All we have to do now is solve this equation for z.

1 - 3z = ln ( 1/5 )

-3z = -1 + ln ( 1/5 )

z = - 1/3 ( -1 + ln ( 1 /5) ) = 0.8698126372

Now that we've seen a equations where the variable only appears in the exponent we need to see an example along with variables both in the exponent & out of it.