The Algebra Teacher\'s Guide to Reteaching Essential Concepts and Skills

Teaching Notes 8.19: Writing Logarithmic Equations as

Exponential Equations

When students are given a logarithmic equation, they can express it as an exponential equation.

They must understand where to substitute they-value, the base, and thex-value.

1. Explain that an exponential function has a numerical base and an exponent that is a variable.
   f(x)=bxis the general form.


2. Explain that the logarithmic function,g(x)=logbx, is the inverse of the exponential func-
   tion.


3. Explain that they-values (the range) of any function arex-values (the domain) of its inverse.
   Depending on the abilities of your students, you may wish to illustrate this concept by using
   two functions your students are familiar with, for example, the squaring function,f(x)=x^2 ,
   and its inverse, the square root function,g(x)=

√

x.f(2)=4, thereforeg(4)=2;f(6)=36,

thereforeg(36)=6.

4. Expand this concept to the exponential function,f(x)=bx, and its inverse,g(x)=logbx.
   For example, if the base is 3,f(2)= 32 or 9, thereforeg(9)=log 39 =2 because 3^2 =9.
   Ifthebaseis3,f(4)= 34 or 81, thereforeg(81)=log 381 =4 because 3^4 =81. Be sure to
   point out to your students that the value of the logarithmic function is the same as the expo-
   nent in the exponential function.


5. Review the information and examples on theworksheet with your students. Make sure that
   they understand all the steps of the examples.

EXTRA HELP:

The subscript in the logarithmic function is the base in the exponential function.

ANSWER KEY:

(1) 23 = 8 (2) 25

1

(^2) = 5 (3) 72 = 49 (4) 16
3
(^2) = 64 (5) 8 −
2
(^3) =^1
4

(Challenge)log√ 55 = 2

310 THE ALGEBRA TEACHER’S GUIDE