Solving Logarithmic Equations

We need to solve log equations to find the

y intercept. We’ll use the log properties to help us do that.

5𝑥−3=32𝑥−1Type 1:

5𝑥−3=32𝑥−1Take the Log or In of both sides.

𝑙𝑜𝑔5𝑥− 3=𝑙𝑜𝑔32𝑥−1Use the power property to bring the

exponents down.(𝑥−3)𝑙𝑜𝑔5❑=(2𝑥−1) 𝑙𝑜𝑔3❑

Use the power property to bring the exponents down.

(𝑥−3)𝑙𝑜𝑔5❑=(2𝑥−1) 𝑙𝑜𝑔3❑

Remember log5 and log3 are just decimals so we can either distribute them through and

then collect like terms, etc. OR….

(𝑥−3)𝑙𝑜𝑔5❑=(2𝑥−1) 𝑙𝑜𝑔3❑Or divide both sides by one of them

𝑙𝑜𝑔5❑𝑙𝑜𝑔5❑

(𝑥−3)❑=(2 𝑥−1 )( .𝟔𝟖𝟐𝟔)Distribute, collect like-terms, etc.

x – 3 = 1.36x - .6826-1.36x + 3 -1.36x +3

x – 3 = 1.36x - .6826x =-6.437

Your turn…..

3𝑥−3=92 𝑥+1

Your turn…..

𝑒𝑥−3=5

Type 2:𝑙𝑜𝑔6 (5 𝑥−3 )=𝑙𝑜𝑔66 𝑥Notice:1. log with same base on both sides.2. logs are ALONE. Nothing is multiplied or added on.For example:𝟐 𝒍𝒐𝒈𝟔 (𝟓 𝒙−𝟑 )=𝒍𝒐𝒈𝟔𝟔 𝒙𝟐 𝒍𝒐𝒈𝟔 (𝟓 𝒙−𝟑 )=𝒍𝒐𝒈𝟔𝟔 𝒙+𝟏𝟎

Type 2:𝑙𝑜𝑔6 (5 𝑥−3 )=𝑙𝑜𝑔66 𝑥

When we have THIS situation and

this situation ONLY the logs can be cancelled.

Type 2:𝑙𝑜𝑔6 (5 𝑥−3 )=𝑙𝑜𝑔66 𝑥

5x - 3And then

it’s easy to solve!

Type 3:𝑙𝑜𝑔6 (5 𝑥−3 )=12

Notice: ONLY one “log”So MUST rewrite into

exponential form

Type 3:𝑙𝑜𝑔6 (5 𝑥−3 )=12

𝟔𝟏𝟐=𝟓 𝒙−𝟑And then

it’s easy to solve!

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