Answer:
Explanation:
You want to evaluate the following expressions using log tables.
- (0.008794÷0.0869)²
- 0.809²×√0.0657
- 0.7805¹⁰
Rules of logarithms
The rules of logarithms we will use here are ...
log(ab) = log(a) +log(b)
log(10^a) = a
log(a/b) = log(a) -log(b)
log(a^b) = b·log(a)
log(√a) = log(a)/2
Table
We don't have a table of logarithms, so we used a calculator to compute them. Our short table is shown in the first line of the first attachment. These 5-digit logarithms are adequate for the 3-digit results we want.
Approach
A table of logarithms gives you the log of numbers between 1 and 10. When a number is written in scientific notation, the coefficient is a number between 1 and 10, so it works well to start by writing these problems in scientific notation. The powers of 10 can be dealt with separately, so we only need to use logarithms for the required math on the coefficients. The appropriate power of 10 is then applied after we take the antilog of the result.
Division
(0.008794 ÷ 0.0869)² = ((8.794×10^-3)/(8.69×10^-2))²
= (8.794/8.69)²×(10^(2(-3-(-2)))
And the log is ...
2(log(8.794) -log(8.69)) -2 = 2(0.94419 -0.93902) -2 = 0.01034 -2
The antilog is ...
(10^0.01034)×10^-2) ≈ 1.02×10^-2 = 0.0102
(0.008794 ÷ 0.0869)² ≈ 0.0102
Root
0.809²·√0.0657 = (8.09×10^-1)²·√(6.57×10^-2)
= (8.09²×√6.57)×10^(-2-1)
And the log is ...
2·log(8.09) +log(6.57)/2 -3 = 2(0.90795) +0.81757/2 -3 = 2.224685 -3
≈ 0.22469 -1
The antilog is ...
(10^0.22469)×10^-1 ≈ 1.68×10^-1 = 0.168
0.809²·√0.0657 ≈ 0.168
Power
0.7805¹⁰ = (7.805×10^-1)¹⁰ = 7.805¹⁰×10^-10
And the log is ...
10·log(7.805) -10 = 10(0.89237) -10 = 8.9237 -10 = 0.9237 -2
The antilog is ...
(10^0.9237)×10^-2 ≈ 8.39×10^-2 = 0.0839
0.7805¹⁰ ≈ 0.0839
__
Additional comment
The second attachment shows the calculator evaluation of each expression. The results agree to 3 significant figures, as we expect.
<95141404393>