Solve the following equations:
a) 5^(2x+1) = 48 b) log6(x+41)-log4(x+1)=2
52x+1 = 48
Take logs of both sides
log(52x+1) = log(48)
Use the rule: log(AB) = B·log(A) on the left side
(2x+1)log(5) = log(48)
Divide both sides by log(5)
(2x+1)log(5) log(48)
———————————— = —————————
log(5) log(5)
1
(2x+1)log(5) log(48)
———————————— = —————————
log(5) log(5)
1
log(48)
2x + 1 = —————————
log(5)
log(48)
2x = ————————— - 1
log(5)
log(48) log(5)
2x = ————————— - —————————
log(5) log(5)
log(48) - log(5)
2x = ——————————————————
log(5)
Divide both sides by 2:
log(48) - log(5)
x = ——————————————————
2·log(5)
1.681241237 - 0.6989700043
x = ————————————————————————————
2·(0.6989700043)
0.982271233
x = —————————————
1.397940009
x = .7026562134
===============================================
b)
log6(x+41) - log4(x+1) = 2
Use rule:
log(A) - log(B) = log(A/B) on left side
æ6(x+41)ö
log ç———————÷ = 2
è 4(x+1)ø
Use the rule: log(A) = B is equivalent to
the equation A = 10B
6(x+41)
———————— = 102
4(x+1)
3
6(x+41)
———————— = 102
4(x+1)
2
3(x+41)
———————— = 100
2(x+1)
Multiply both sides by 2(x+1)
3(x+41) = 100·2(x+1)
3(x+41) = 200(x+1)
3x + 123 = 200x + 200
-197x = 77
x = -77/197
Edwin McCravy
AnlytcPhil@aol.com