D-10 Solving Log Equations Using Properties Notes

→ Equations in the form of 𝑙𝑜𝑔𝑏(𝑥) = 𝑙𝑜𝑔𝑏(𝑦)

Examples: Solve the equation for x. Round answers to 3 decimal places. a. 𝑙𝑜𝑔13(2𝑥) = 𝑙𝑜𝑔13(𝑥2 − 𝑥 + 2) b. 𝑙𝑜𝑔3(𝑥2 + 3) = 𝑙𝑜𝑔3(52)

c. ln (x + 2) + ln (3x – 2) = 2 ln (2x) d. 𝑙𝑜𝑔3(7𝑥 + 3) − 𝑙𝑜𝑔3(𝑥 + 1) = 𝑙𝑜𝑔3(2𝑥) e. log (x) + log (x – 3) = log (28) f. ln (x - 5) + ln 4 = ln x - ln 2

→ Solving log equations in the form 𝑙𝑜𝑔𝑏(𝑥) = 𝑐

a. 3 + 𝑙𝑜𝑔9(4𝑥) = 5 b. 2 = −3 + ln (𝑥 + 2)

→ Solving log equations in the form 𝑙𝑜𝑔𝑏(𝑥) + 𝑙𝑜𝑔𝑏(𝑦) = 𝑐 or 𝑙𝑜𝑔𝑏(𝑥) − 𝑙𝑜𝑔𝑏(𝑦) = 𝑐

Examples: Solve each equation for x. Round to the nearest hundredth. a. 𝑙𝑜𝑔12(12𝑥) + 𝑙𝑜𝑔12(𝑥 − 1) = 2 b. log (x – 12 ) – log (x – 2 ) = 2

c. log (50x ) = 2 + log( 2x - 3 ) d. 𝑙𝑜𝑔1/4 (1

4𝑥) = −

5

2 − 𝑙𝑜𝑔1/4(𝑥 + 8)

d. e. 𝑙𝑜𝑔2(𝑥) + 𝑙𝑜𝑔2(𝑥 − 2) = 3