Solve for x if log(𝑥 − 1) + log(𝑥 + 1) = 𝑙𝑜𝑔21
Step 1. Apply the Product Rule Law: 𝒍𝒐𝒈𝒙 𝒂 + 𝒍𝒐𝒈𝒙 𝒃 = 𝒍𝒐𝒈𝒙 𝒂𝒃
log(𝑥 − 1) + log(𝑥 + 1) = 𝑙𝑜𝑔21
log(𝑥 − 1) (𝑥 + 1) = 𝑙𝑜𝑔21
Apply the rule of the product of the binomial sum and
difference which is equal to the square of the first term minus
the square of the second term.
log(𝑥 2 − 1) = 𝑙𝑜𝑔21
Step 2. Equate.
log(𝑥 2 − 1) = 𝑙𝑜𝑔21
𝑥 2 − 1 = 21
𝑥 2 − 1 + 1 = 21 + 1
Add 1 to the both sides of equation to eliminate -1.
𝑥 2 = 22
√𝑥 2 = √22
𝒙 = √𝟐𝟐
To check:
log(√22 − 1) + log(√22 + 1) = 𝑙𝑜𝑔21
log(22 − 1) = 𝑙𝑜𝑔21
log 21 = 𝑙𝑜𝑔21
Find the square root to eliminate the square of x.