Exercise 24 & 25: Exponential and Logarithmic Equations I & II

Today, Wednesday, April 20th, we learned how to evaluate exponential equation when the bases of the powers cannot be made to be the same. Also we learned how to logarithmic equations that have the same base. Logarithmic equations with different bases can be solved by using the change of base theorem which we discussed during the sixth period class.

Evaluating Exponential Equations with Dissimilar Bases

Recall:
An example equation is:

(2 to the x) = 16

16 can be written as a power with a base of 2:

(2 to the x) = (2 to the 4)

Because the bases are the same, we can equate the exponents:

x = 4

Although this example was very simple, the same process can be used for any exponential equation that has the same bases.


Now, on to discuss equations that cannot be written with equal bases. I will explain the process while trying to show an example:

(10 to the x) = log 1.574

The first step is to apply logs to both sides of the eqation:

log (10 to the x) = log 1.574

The exponential power law can be applied to the L.H.S.:

x log 10 = log 1.574

Now you must isolate x:

x = (log 1.574)/(log1o)

Simply input it into your calculater:

x = 0.1970

Warning: You can only solve these equations usinga calculator if the log have a base of 10!!!!!!!!!


Evaluating Logarithmic Equations

The Example:

log x + log(x-3) = 1

The first step is to move all the logs to one side of the equation. Following that you must write that expression as a single log using the exponential product law and the exponential quotient law as applicable:

log x(x-3) = 1

The next step is to write the equation in exponential form:

(10 to the 1) = x(x-3)

Solve for x:

10 = (x to the 2) -3x

0 = (x to the 2) -3x -10

0 = (x-5)(x+2)

x = +5, -2

You must cross out any negative answer as you cannot take the log of a negative number.

x = 5

Change the Base of the Log

The change of base theorem is need when you are required to evaluate an expression that is not in base 10. You need not memorize it as it is on your formula sheet.

(log n base b) = (log n)/(log b)

An example:

Evaluate: log 3 base 2

You simple take the log of the power and divide it by the log of the base:

(log 3)/(log 2)

Enter this base 10 log into your calculator:

1.5850

Homework: Exercise 24: 1-11, 13, 15, 16, 18 and Exercise 25. A hand-in assignment was also distributed.

Lastly: I used to the phrase `to the` to shortform the phrase `raised to the power of` due to my inability to use superscripts.