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