Hints for solving problems in Apply 12.2

Problem No. 2 - Write the exponential statement in logarithmic form:

Remember, exponential notation has a base (4), an exponent (5), and a power or product (1024).

The logarithmic form has the base written to the right and below the word log. Also to the right of the word log on the same line is the power or product and after the equal sign is the exponent.

Remember, the logarithm of the base is the exponent needed to rise to the power or product shown.

                          the answer

Problem No. 6 - Write the exponential statement in logarithmic form:

The base is: (b). The exponent is: (x). The power or product is: (36)

                              the answer

Problem No. 10 - Find:      

Change the problem to its exponential form and solve.

                                   x = 2        the answer

Problem No. 16 - What is the inverse of the function?  

     Notice in this problem the y is the exponent and x is the power or product. So to get the inverse you change the position of the x and the y. This is the inverse in log form, but the answer the book wants is in exponential form. So change this to its exponential form.

          Now put it in functional form.          the answer

Problem No. 20 - Write this logarithmic statement in exponential form:

Remember, the number written to the right and a little below the word log is the base. The number just to the right of the word log on the same line is the power or product. The number written by itself on one side of the equal sign is the exponent.

                       the answer

Problem No. 26 - Graph the function:   

There two ways to do this. If is says to graph using the inverse, that is one way and if it says to use the exponential form that is the other.

I will show both ways.

Using the inverse:

               The inverse is:      Start here

        x         y                                 x         y

               -1                                -1                   

        1         0                                  0         1             y = 1

        2         1                                  1         2             y = 2

The inverse was in exponential form. The x and y were exchanged to get the correct x and y for the log equation. If you graph the three order pairs for the log equation, you will get the correct graph.

Using the exponential form:

                The exponential form is:   

                                                                           x         y

                                                                                  -1        

                                                                           1         0            x = 1

                                                                           2          1           x = 2

Notice these ordered pairs are the same as the ones under the log equation above. The graph will naturally be the same.

The following logarithmic properties could be used in the following exercises.

Log of a product

         This says the log of any product is the sum of the logs of the factors.      

Log of a Quotient

     This says the log of any quotient is the difference of the logs. (The numerator minus the denominator)

Log of a Power

          This says when the log of a number or letter has an exponent, the exponent is moved to the front of the word log and multiplied.

Problem No. 30 - Rewrite using the log of a power property:

                  =        the answer

This property will be used the most, as a factor in front of the log is moved to the exponent of the product or power.

Most of the time when solving log equations, you will move the exponent of the product or power down to be a factor in front of the word log.

Problem No. 34 - Rewrite using the log of a quotient property:

            =        the answer

Problem No. 36 - Rewrite using the log of a product property to get an expression with two terms:

            =           the answer

Problem No. 38 - Rewrite as a single logarithm:

              =        the answer

Remember, if it is the sum of logs, it will be the product. If it is the difference between logs, is will be a quotient.

Problem No. 42 - Rewrite as a single logarithm:

              =        the answer

Problem No. 44 - Rewrite as a single logarithm:

The first step is to move all factors in front of the word log to become exponents of the power or product.

Then if there is a positive sign in front of the term is will be a product and if there is a negative sign in front of the term is will be a quotient.

          =        =        the answer

Problem No. 48 - Simplify:    

If you read the exercise there was a property that says if the exponent was a log and had the same base as the original base, then the answer is the product or power of the exponent, which is 3. the answer

Here is another way to solve the problem.

     Change to log form:      Now when the logs and the bases are equal the products or powers are equal. Drop the word log and you get x = 3. the answer

Problem No. 51 - Simplify:    

There was a property in book concerning this also. If the base was the same as the product or power, then the exponent is the answer (-2).

Here is another way to solve the problem.

     Change to exponent form:      Now if the bases are the same the exponents are equal. x = -2 the answer

Problem No. 56 - Rewrite using the properties of logarithms to get an expression with three terms.

     Rewrite the expression with positive and negative signs between and at the end bring all the exponents of the products or powers down as a factor in front of the word log.

         =      the answer