Common Logarithms (Base 10)

Show your work for each problem.

1.
What is $\log 10\,000$ ?
1. a)1000
2. b)10000
3. c)3
4. d)4
Answer: 4
$10\,000 = 10^4$ , so $\log 10\,000 = 4$ . The log counts the zeros (factors of 10), not all the digits.
2.
What is $\log 1\,000\,000$ ?
Answer: 6
$1\,000\,000 = 10^6$ (six zeros), so $\log 1\,000\,000 = 6$ .
3.
What is $\log 0.01$ ?
1. a)-2
2. b)2
3. c)0.01
4. d)-0.01
Answer: -2
$0.01 = \dfrac{1}{100} = 10^{-2}$ , so $\log 0.01 = -2$ . Numbers between 0 and 1 have negative logs.
4.
What is $\log 1$ ?
Answer: 0
$10^0 = 1$ , so $\log 1 = 0$ . Do not mix this up with $\log 0$ , which does not exist.
5.
Find $\log 0.001$ .
Answer: -3
1. $0.001 = \dfrac{1}{1\,000}$ . Write $1\,000 = 10^n$ . What is $n$ ? 3
2. So $0.001 = 10^k$ . What is $k$ ? -3
3. What is $\log 0.001$ ? -3
6.
Find $\log 0.1$ , starting from $\log 10$ .
Answer: -1
1. What is $\log 10$ ? 1
2. Write $0.1 = \dfrac{1}{10}$ as $10^n$ . What is $n$ ? -1
3. What is $\log 0.1$ ? -1
7.
Between which two whole numbers is $\log 50$ ?
1. a)2 and 3
2. b)0 and 1
3. c)5 and 6
4. d)1 and 2
Answer: 1 and 2
$10 < 50 < 100$ , so $\log 10 < \log 50 < \log 100$ , that is $1 < \log 50 < 2$ . The calculator gives $\log 50 \approx 1.699$ .
8.
Estimate $\log 700$ between two whole numbers, then find it with the calculator to 3 decimal places.
Answer: 2.845
1. $700$ lies between $100$ and $1\,000$ . What is $\log 100$ ? 2
2. What is $\log 1\,000$ ? 3
3. Type log(700) in the calculator. What is $\log 700$ to 3 decimal places? 2.845
9.
Use the calculator: what is $\log 250$ , rounded to 2 decimal places?
Answer: 2.40
$\log 250 = 2.39794\ldots \approx 2.40$ . It lies between $\log 100 = 2$ and $\log 1\,000 = 3$ , as expected.
10.
A positive number $x$ has $\log x = 3$ . Find $x$ .
Answer: 1000
1. Rewrite $\log x = 3$ as $b^3 = x$ . What is the base $b$ ? 10
2. What is $x = 10^3$ ? 1000
3. Check: what is $\log 1\,000$ ? 3
11.
Which value is closest to $\log 2\,000$ ?
1. a)0.301
2. b)4.301
3. c)3.301
4. d)2.301
Answer: 3.301
$1\,000 < 2\,000 < 10\,000$ , so $3 < \log 2\,000 < 4$ . Only $3.301$ is in that range, and the calculator confirms $\log 2\,000 \approx 3.301$ .
12.
Estimate $\log 0.05$ between two whole numbers, then find it with the calculator to 3 decimal places.
Answer: -1.301
1. $0.05$ lies between $0.01$ and $0.1$ . What is $\log 0.01$ ? -2
2. What is $\log 0.1$ ? -1
3. Type log(0.05) in the calculator. What is $\log 0.05$ to 3 decimal places? -1.301
13.
A positive number $x$ has $\log x = -2$ . What is $x$ ? Write it as a decimal.
Answer: 0.01
$\log x = -2$ means $x = 10^{-2} = \dfrac{1}{100} = 0.01$ .
14.
Write $62\,000$ in scientific notation, find the whole-number part of $\log 62\,000$ , then find $\log 62\,000$ to 3 decimal places.
Answer: 4.792
1. $62\,000 = 6.2 \times 10^n$ . What is $n$ ? 4
2. So $4 \le \log 62\,000 < k$ . What is the whole number $k$ ? 5
3. Type log(62000) in the calculator. What is $\log 62\,000$ to 3 decimal places? 4.792
15.
A whole number $N$ has $\log N \approx 5.3$ . How many digits does $N$ have?
1. a)5
2. b)53
3. c)3
4. d)6
Answer: 6
$5 \le 5.3 < 6$ , so $10^5 \le N < 10^6$ , that is $100\,000 \le N < 1\,000\,000$ . These numbers have 6 digits ( $N \approx 10^{5.3} \approx 199\,526$ ).
16.
A positive number $x$ has $\log x = 2.5$ . Find $x$ , rounded to the nearest whole number.
Answer: 316
1. $x = 10^{2.5}$ lies between $10^2$ and $10^3$ . What is $10^2$ ? 100
2. What is $10^3$ ? 1000
3. Type 10^2.5 in the calculator. What is $x$ to the nearest whole number? 316
17.
A whole number $N$ has $\log N \approx 7.48$ . How many digits does $N$ have?
Answer: 8
$7 \le 7.48 < 8$ , so $10^7 \le N < 10^8$ : $N$ is between $10\,000\,000$ and $99\,999\,999$ , so it has 8 digits. In general, the number of digits is the whole-number part of the log plus 1.
18.
Use common logs to find how many times larger $100\,000$ is than $100$ .
Answer: 1000
1. What is $\log 100\,000$ ? 5
2. What is $\log 100$ ? 2
3. By how much do the two logs differ? 3
4. A difference of 3 means a factor of $10^3$ . How many times larger is $100\,000$ ? 1000