Logs of Powers of 10

Find $\log 100\,000$ and $\log 1$ without a calculator.

Ask the question for 100 000: 10 to which power gives $100\,000$? = $10^n = 100\,000$

Count the zeros: $100\,000$ is a 1 followed by 5 zeros, so $100\,000 = 10^5$ = $\log 100\,000 = 5$

Ask the question for 1: Any nonzero number to the power 0 is 1: $10^0 = 1$ = $\log 1 = 0$

Answer: $\log 100\,000 = 5$ and $\log 1 = 0$

Logs of Small Decimals

Find $\log 0.001$.

Write the decimal as a fraction: $0.001 = \dfrac{1}{1\,000}$ = $\dfrac{1}{10^3}$

Use a negative exponent: $\dfrac{1}{10^3} = 10^{-3}$ = $0.001 = 10^{-3}$

Read off the exponent: $\log(10^{-3}) = -3$ = $\log 0.001 = -3$

Answer: $\log 0.001 = -3$. Numbers between 0 and 1 have negative logs.

Estimating log 50

Between which two whole numbers is $\log 50$? Then find it with a calculator.

Find the powers of 10 around 50: $10 < 50 < 100$, that is $10^1 < 50 < 10^2$ = Neighbours: $10$ and $100$

Take the logs of the neighbours: $\log 10 = 1$ and $\log 100 = 2$ = $1 < \log 50 < 2$

Use the LOG key: Type log(50) = $\log 50 \approx 1.699$

Check the size: Halfway on a log scale is $10^{1.5} \approx 31.6$, not 55. Since $50 > 31.6$, $\log 50$ is above $1.5$ = $1.699$ makes sense

Answer: $1 < \log 50 < 2$, and $\log 50 \approx 1.699$

Logs and Scientific Notation

Write $3\,000$ in scientific notation, predict the whole-number part of $\log 3\,000$, then check with a calculator.

Write it in scientific notation: $3\,000 = 3 \times 10^3$ = Exponent: $3$

Find the powers of 10 around it: $10^3 = 1\,000 \le 3\,000 < 10\,000 = 10^4$ = $3 \le \log 3\,000 < 4$

Use the LOG key: Type log(3000) = $\log 3\,000 \approx 3.477$

Compare: The whole-number part $3$ is the exponent from scientific notation = The order of magnitude is $3$

Answer: $\log 3\,000 \approx 3.477$: whole-number part $3$, the same as the exponent in $3 \times 10^3$

From the Log Back to the Number

A positive number $x$ has $\log x = 2.5$. Between which powers of 10 is $x$, and what is $x$ (to 2 decimal places)?

Rewrite with powers of 10: $\log x = 2.5$ means $x = 10^{2.5}$ = $x = 10^{2.5}$

Place the exponent: $2 < 2.5 < 3$, so $10^2 < x < 10^3$ = $100 < x < 1\,000$

Use the calculator: Type 10^2.5 = $x \approx 316.23$

Answer: $x = 10^{2.5} \approx 316.23$, a number between $100$ and $1\,000$ (3 digits)

Mistake: Thinking $\log 50 = 1.5$ because 50 is about halfway between 10 and 100

Why: A log scale is not evenly spaced. Each step of 1 in the log multiplies the number by 10, so the halfway point between 10 and 100 is $10^{1.5} \approx 31.6$.

Correct: $\log 50 \approx 1.699$. Estimate the two whole numbers first ($1 < \log 50 < 2$), then use the calculator.

Mistake: Writing $\log 0.01 = 2$ or saying the log of a decimal does not exist

Why: $0.01 = 10^{-2}$, so the exponent is negative. Logs of numbers between 0 and 1 exist, they are just negative.

Correct: $\log 0.01 = -2$, $\log 0.1 = -1$, $\log 0.001 = -3$.

Mistake: Writing $\log 0 = 0$ or $\log(-100) = -2$

Why: $10^n$ is positive for every exponent $n$. No power of 10 equals 0 or a negative number.

Correct: $\log 0$ and $\log(-100)$ are undefined. The calculator shows an error. Remember: $\log 1 = 0$, not $\log 0$.

Mistake: Counting all the digits: $\log 1\,000 = 4$

Why: $1\,000$ has 4 digits but only 3 zeros. The log counts the factors of 10, not the digits.

Correct: $\log 1\,000 = 3$ because $10^3 = 1\,000$. A whole number with $d$ digits has a log between $d - 1$ and $d$.

Orders of Magnitude

Scientists compare huge and tiny quantities by their order of magnitude: the whole-number part of their common logarithm.

A human hair is about $0.0001$ m thick, and $\log 0.0001 = -4$. The Earth is about $10\,000\,000$ m across, and $\log 10\,000\,000 = 7$. The difference of 11 orders of magnitude means the Earth is about $10^{11}$ times wider.

pH in Chemistry

The pH of a solution is $\text{pH} = -\log[H^+]$, where $[H^+]$ is the hydrogen ion concentration in mol/L. Lower pH means more acidic.

Pure water has $[H^+] = 0.0000001 = 10^{-7}$, so $\text{pH} = -(-7) = 7$ (neutral).

Decibels and Sound

Sound level in decibels is $L = 10 \log\left(\dfrac{I}{I_0}\right)$, where $\dfrac{I}{I_0}$ says how many times stronger a sound is than the quietest sound we can hear.

A whisper is about $100$ times the threshold: $L = 10 \log 100 = 10 \times 2 = 20$ dB.

Q: Why is base 10 called the common logarithm?

A: Our number system is built on 10, so base 10 logs were the ones printed in log tables for centuries and the ones on every scientific calculator (the LOG key). Because they are used so often, the base is simply left out: $\log x$ means $\log_{10} x$.

Q: How does the calculator find $\log 50$?

A: It uses a fast numeric method to find the exponent $n$ with $10^n = 50$. You can check its answer: $10^{1.699} \approx 50$. The estimate $1 < \log 50 < 2$ tells you if you pressed the right keys.

Q: Why are logs of decimals like $0.01$ negative?

A: Numbers between 0 and 1 are reached with negative exponents: $0.01 = \dfrac{1}{100} = 10^{-2}$. So $\log 0.01 = -2$. Only the argument must be positive, the log itself can be any real number.

Q: Is $\log$ on my calculator the same as $\ln$?

A: No. LOG uses base 10, while LN uses a different base, the number $e \approx 2.718$. For example $\log 100 = 2$, but $\ln 100 \approx 4.605$. The lesson Natural Logarithms (ln) covers that key.