M6L5SOLVING EXPONENTIALS USING LOGS

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) 𝟑𝟒𝒙−𝟏 = 𝟖𝟏 (81 is the same as 34)

𝟑𝟒𝒙−𝟏 = 𝟑𝟒

Since 3 to a power equals 3 to the 4th, those powers must be equal…

𝟒𝒙 − 𝟏 = 𝟒𝟒𝒙 = 𝟓

𝒙 =𝟓

𝟒

Let’s try that same idea on this problem.

2) 𝟐𝟒𝒙−𝟏 = 𝟖𝒙(8 is the same as 23)

𝟐𝟒𝒙−𝟏 = (𝟐𝟑)𝒙

𝟐𝟒𝒙−𝟏 = 𝟐𝟑𝒙

Again since our bases are the same, our powers must be equivalent…

𝟒𝒙 − 𝟏 = 𝟑𝒙−𝟏 = −𝟏𝐱

𝒙 = 𝟏

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) 𝟑𝟒𝒙 = 𝟓𝟎

If we take the log of each side, the power property allows the 4x to move out front.

𝒍𝒐𝒈(𝟑𝟒𝒙) = 𝒍𝒐𝒈 𝟓𝟎𝟒𝒙 ∗ 𝒍𝒐𝒈(𝟑) = 𝒍𝒐𝒈 𝟓𝟎

𝟒𝒙 =𝒍𝒐𝒈(𝟓𝟎)

𝒍𝒐𝒈(𝟑)

𝒙 =𝒍𝒐𝒈(𝟓𝟎)

𝒍𝒐𝒈(𝟑)÷ 𝟒

𝒙 = 𝟎. 𝟖𝟗𝟎𝟐𝟏𝟗𝟏𝟗𝟖𝟖

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) 𝟏𝟑𝟎𝟑𝒙+𝟏 = 𝟏𝟕𝟓 take log of both sides

𝒍𝒐𝒈(𝟏𝟑𝟎𝟑𝒙+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

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: 𝟏𝟑𝟎𝟑𝒙+𝟏 = 𝟏𝟕𝟓

To check, graph y1= 𝟏𝟑𝟎𝟑𝒙+𝟏

y2= 𝟏𝟕𝟓

*Please note our answer algebraically was 0.0203560637 whereas the graph rounded to 0.2.

Here are some great website resources with examples:

• Regents Prep examples:http://www.regentsprep.org/regents/math/algtrig/ate8/exponentialequations.htm

• Khan Academy video:https://www.khanacademy.org/math/algebra2/exponential-and-logarithmic-functions/solving-exponential-equations-with-logarithms/v/exponential-equation