8e)++Using+logs+to+solve+exponential+equations

Welcome to Section 8e, Using Logs to Solve Exponential Equations!!!!!! We (Hallie, Rachel, Kathryn, Mr. Wiggglzwerth, and me, Elizabeth) hope this site is useful in teaching you everything you absolutely need to know about using logs to solve exponential equations. We hope that our examples and links help you so much that, say you "forgot" that logs were on the final, you could come here and easily understand the topic. So, without further ado: using logs to solve exponential equations!

As you know from previous sections, a logarithm "undos" an exponential equation. A logarithm is basically an exponential equation flipped around. For example, imagine the textbook has a problem like:

You can use a log to simplify this problem. Since 2 is the base, x is the exponent, and 16 is the result, you can transform this into:

From there, it is easy to use the change of base formula and find that the log of 16 divided by the log of 2 equals 4.

Now that you know how to convert an exponential equation to log form and back again and you know how to expand and condense, I (Hallie) am going to introduce you to using logs to solve exponential equations. Exponential equations are equations in which variable expressions occur as exponents. There are two properties that are essential when solving equations.

If b is a positive number other than 1, then if and only if x=y. For Example: If, then x=5, then.
 * Property of equality for exponential equations:**

If b,x, and y are positive numbers with, then if and only if x=y.
 * The property of Equality for Logarithmic Equations:**

For Example: If, then x=7. If x=7, then.

If these properties are confusing use the websites, tutorial, and practice problems to help you to understand. We hope that this wiki will help you study for the the Algebra 2 final. Good Luck!


 * Here are some sample problems to help you further understand this topic:**

** Solve 5//x// = 212 Since 212  is not a power of 5 , then I will have to use logs to solve this equation. I could take base- 5  log of each side, solve, and then apply the change-of-base formula, but I think I'd rather just use the natural log in the first place:  ** 5//x// = 212 //ln//(5//x//) = //ln//(212) //xln//(5) = //ln//(212) ** //x// = //ln//(212)///ln//(5) **

** Solve 102//x// = 52. **

Since 52  is not a power of 10 <span style="color: rgb(0, 0, 0); font-family: Arial;">, I will have to use logs to solve this. In this particular instance, since the base is <span style="color: rgb(0, 0, 0); font-family: 'Times New Roman';">10 <span style="color: rgb(0, 0, 0); font-family: Arial;"> and since base- <span style="color: rgb(0, 0, 0); font-family: 'Times New Roman';">10 <span style="color: rgb(0, 0, 0); font-family: Arial;"> logs can be done on the calculator, I will use the common log instead of the natural log to solve this equation: <span style="color: rgb(0, 0, 0); font-family: 'Times New Roman';">102//x// = 52 //log//(102//x//) = //log//(52) 2//xlog//(10) = //log//(52) 2//x//(1) = //log//(52) 2//x// = //log//(52) <span style="color: rgb(128, 0, 128); font-family: 'Times New Roman';">**//x// = //log//(52)/2**

Below are some links to useful websites, several sample problems, and even a tutorial! Have fun! [] [|http://www.sosmath.com/algebra/logs/log4/log46/log46.html]