L3 Exponential and Logarithmic Functions PDF

**By the end of this section, I should be able to\...** - - - - ### A. The Inverse of Exponential Functions **Recall:** an exponential function is a function that has a variable in the exponent such as [*y* = 2^*x*^.]{.math.inline} How can we solve an exponential equation such as [100 =...

**By the end of this section, I should be able to\...** - - - - ### A. The Inverse of Exponential Functions **Recall:** an exponential function is a function that has a variable in the exponent such as [*y* = 2^*x*^.]{.math.inline} How can we solve an exponential equation such as [100 = 2^*x*^]{.math.inline}? It would be great if we had a function that could **undo** the exponential. In other words, an inverse of the exponential. Turns out we do! It's called a logarithm. Complete the table below to summarize the characteristics of exponential and logarithmic functions: **Characteristic** **Exponential (base 2)** **Logarithmic (base 2)** ---------------------- ------------------------------- ------------------------------- Parent equation [*y* = 2^*x*^]{.math.inline} [*x* = 2^*y*^]{.math.inline} Sketch Domain Range Y-intercept X-intercept Asymptotes? Interval of increase Interval of decrease ### B. Logarithmic Notation The inverse of the exponential [*y* = *b*^*x*^]{.math.inline} is the logarithmic [*x* = *b*^*y*^]{.math.inline}. The problem with this form is that it is not **explicit** (y = \_\_\_\_). To fix this, mathematicians created notation that looks like this: +-----------------------------------+-----------------------------------+ | [*y* = log~*b*~*x*]{.math | *where\...* | |.inline} | | | | - - - | +-----------------------------------+-----------------------------------+ What restrictions does [log~*b*~*x*]{.math.inline} have? **Example 1**: Change to exponential form. a. **Example 2**: Express as a logarithm. a. ### C. Evaluating Logarithms In order to evaluate logarithms, convert them to their exponential form. **[Example 1]**: Evaluate. a. d. g. j. **In general:** the following are useful to remember\... --------------------------------------------------------------------------------------------------------------------------------- [log~*b*~1=]{.math.inline} [log~*b*~*b*=]{.math.inline} [log~*b*~*b*^*x*^=]{.math.inline} [*b*^log~*b*~*x*^=]{.math.inline} --------------------------------------------------------------------------------------------------------------------------------- **Practice:** 8.1 \#7,9,10 and 8.3 \#1-6, 9-10