Properties of Logarithms
Check for Understanding – 3103.3.16 – Prove basic properties of logarithms using properties of exponents and apply those
properties to solve problems.
Check for Understanding – 3103.3.17 – Know that the logarithm and exponential functions are inverses and use this
information to solve real-world problems.
Since logarithms are
exponents, the properties of
logarithms are similar to the
properties of exponents.
Product Property
logb mn = logb m + logb n
Quotient Property
logb m = logb m – logb n
n
Power Property
logb mp = p logb m
m > 0, n > 0, b > 0, b ≠ 1
Use log2 3 ≈ 1.5850, log2 5 ≈ 2.3219, and log2 7 ≈ 2.8074
to approximate the value of each expression.
1. log2 35
log2 7 ∙ 5
log2 7 + log2 5
2.8074 + 2.3219
5.1293
Use log2 3 ≈ 1.5850, log2 5 ≈ 2.3219, and log2 7 ≈ 2.8074
to approximate the value of each expression.
2.
log2 45
log2 32 ∙ 5
log2 32 + log2 5
2log2 3 + log2 5
2(1.5850) + 2.3219
5.4919
Use log2 3 ≈ 1.5850, log2 5 ≈ 2.3219, and log2 7 ≈ 2.8074
to approximate the value of each expression.
3. log2 4.2
log2 (3 ∙ 7) ÷ 5
log2 3 + log2 7 – log2 5
1.5850 + 2.8074 – 2.3219
2.0705
Solve each equation. Check your solutions.
4. log5 2x – log5 3 = log5 8
2x
log5
= log5 8
3
2x
3
=8
2x = 24
x = 12
Solve each equation. Check your solutions.
5. log2 (x + 1) + log2 5 = log2 80 – log2 4
log2 5(x + 1)= log2 20
5x + 5 = 20
5x = 15
x=3
Solve each equation. Check your solutions.
6. 3log2 x – 2log2 5x = 2
2
3
100x = x
log2 x3 – log2 (5x)2 = 2 0 = x3 – 100x2
x3
log2 25 x 2
=2
0 = x (x – 100)
2
22 =
x3
25 x 2
0 = x2
0 = x – 100
4=
x3
25 x 2
x=0
x = 100
Solve each equation. Check your solutions.
7. ½ log6 25 + log6 x = log6 20
8. log7 x + 2log7 x – log7 3 = log7 72
Solve each equation. Check your solutions.
7. ½ log6 25 + log6 x = log6 20
4
8. log7 x + 2log7 x – log7 3 = log7 72
6