Logarithms
Laws of Logatithms:
ln x
ln a , loge
ln x
ln e
1.
if loga x = y, then ay = x
2.
loga x =
3.
ln (ex ) = x, ln e = 1
4.
if ln x = y, then ey = x
5.
loga (xy) = loga x + loga y
6.
loga
7.
loga xr = r loga x
Domain of y = loga x : {x|x ≥ 0}; Range: {y|y ∈ R}
x
y
= ln x
= loga x − loga y
1. Solve the following equations for x using the laws of logarithms.
ln x = −1
a.
ln x = 2
b.
c.
ln x = 5
d. 13x = 6
e.
7 = 2x
f.
9x = 4
g.
e5−3x = 10
h.
(2 − ln x) ln x = 0
i.
2 ln x = 1
j.
e−x = 5
k. e2x+3 − 7 = 0
l.
ln (5 − 2x) = −3
2x−5 = 3
n.
ln x + ln (x − 1) = 1
m.
x=
2. Using Laws of Logarithms, simplify the following expressions.
a.
logb x − logb y =
b.
logb u + logb v =
c.
loga y 2 + loga x3 =
d.
loga y 3 + loga x4 =
e.
ln e =
f.
ln a +
h.
ln x + a ln y − b ln z =
g. 2 ln 4 − ln 2 =
1
2
ln b =
3. Find the value of each expression.
1
36
a.
log2 64 =
b.
log6
c.
log8 2 =
d.
√ ln e 2 =
e.
log10 1.25 + log10 80 =
f.
log5 10 + log5 20 − 3 log5 2 =
h.
e3 ln 2 =
g. 2log2 3+log2 5 =
=