M6L5 Solving Exponentials using Logs
- 1. M6L5 SOLVING EXPONENTIALS USING LOGS
- 2. When you’re given an equation to solve where you could write both sides as the same base to a power… use that to help you solve. 1) 𝟑 𝟒𝒙−𝟏 = 𝟖𝟏 (81 is the same as 34) 𝟑 𝟒𝒙−𝟏 = 𝟑 𝟒 Since 3 to a power equals 3 to the 4th, those powers must be equal… 𝟒𝒙 − 𝟏 = 𝟒 𝟒𝒙 = 𝟓 𝒙 = 𝟓 𝟒
- 3. Let’s try that same idea on this problem. 2) 𝟐 𝟒𝒙−𝟏 = 𝟖 𝒙 (8 is the same as 23) 𝟐 𝟒𝒙−𝟏 = (𝟐 𝟑 ) 𝒙 𝟐 𝟒𝒙−𝟏 = 𝟐 𝟑𝒙 Again since our bases are the same, our powers must be equivalent… 𝟒𝒙 − 𝟏 = 𝟑𝒙 −𝟏 = −𝟏𝐱 𝒙 = 𝟏
- 4. Now let’s try one where we cannot rewrite the bases to be the same. (50 cannot easily be represented as 3 to a power.) 3) 𝟑 𝟒𝒙 = 𝟓𝟎 If we take the log of each side, the power property allows the 4x to move out front. 𝒍𝒐𝒈(𝟑 𝟒𝒙 ) = 𝒍𝒐𝒈 𝟓𝟎 𝟒𝒙 ∗ 𝒍𝒐𝒈(𝟑) = 𝒍𝒐𝒈 𝟓𝟎 𝟒𝒙 = 𝒍𝒐𝒈(𝟓𝟎) 𝒍𝒐𝒈(𝟑) 𝒙 = 𝒍𝒐𝒈(𝟓𝟎) 𝒍𝒐𝒈(𝟑) ÷ 𝟒 𝒙 = 𝟎. 𝟖𝟗𝟎𝟐𝟏𝟗𝟏𝟗𝟖𝟖
- 5. Here’s another one where we cannot rewrite the bases to be the same. (175 cannot easily be represented as 130 to a power.) 4) 𝟏𝟑𝟎 𝟑𝒙+𝟏 = 𝟏𝟕𝟓 take log of both sides 𝒍𝒐𝒈(𝟏𝟑𝟎 𝟑𝒙+𝟏 ) = 𝒍𝒐𝒈 𝟏𝟕𝟓 now let’s use the power property (𝟑𝒙 + 𝟏) ∗ 𝒍𝒐𝒈(𝟏𝟑𝟎) = 𝒍𝒐𝒈 𝟏𝟕𝟓 𝟑𝒙 + 𝟏 = 𝒍𝒐𝒈(𝟏𝟕𝟓) 𝒍𝒐𝒈(𝟏𝟑𝟎) 𝟑𝒙 = 𝒍𝒐𝒈(𝟏𝟕𝟓) 𝒍𝒐𝒈(𝟏𝟑𝟎) − 𝟏 𝒙 = 𝒍𝒐𝒈 𝟏𝟕𝟓 𝒍𝒐𝒈 𝟏𝟑𝟎 − 𝟏 ÷ 𝟑 𝒙 = 𝟎. 𝟎𝟐𝟎𝟑𝟓𝟔𝟎𝟔𝟑𝟕
- 6. Remember, you can always check* your answer by graphing. Graph the left side in y1 and the right side in y2. Let’s revisit question 4: 𝟏𝟑𝟎 𝟑𝒙+𝟏 = 𝟏𝟕𝟓 To check, graph y1= 𝟏𝟑𝟎 𝟑𝒙+𝟏 y2= 𝟏𝟕𝟓 *Please note our answer algebraically was 0.0203560637 whereas the graph rounded to 0.2.
- 7. Here are some great website resources with examples: • Regents Prep examples: http://www.regentsprep.org/regents/math/algtrig/a te8/exponentialequations.htm • Khan Academy video: https://www.khanacademy.org/math/algebra2/expon ential-and-logarithmic-functions/solving-exponential- equations-with-logarithms/v/exponential-equation