Chapter 2: Logarithms

Explain · page 27: 3. Find the value of x in each of the following: Gi) Fora number less than 1:

Core idea – The common logarithm of any number smaller than 1 is a negative number because it tells you how many times you must divide 10 to get that tiny number.

What to remember

• $\displaystyle \log_{10}10^{k}=k$ for any integer $k$ (definition of logarithm).
• Numbers less than 1 are powers of 10 with a negative exponent:

$0.1=10^{-1},\;0.01=10^{-2},\;0.001=10^{-3},\dots$

• Therefore $\displaystyle \log_{10}(0.1)=-1,\;\log_{10}(0.01)=-2,\;\log_{10}(0.001)=-3$, etc.
• To find $x$ in an equation like $\log_{10}(\text{number})=x$, rewrite the number as $10^{\text{(something)}}$ and read off the exponent.

Everyday example (Pakistan) – When you look at a cricket scorecard, the run‑rate might be written as “0.75 runs per ball”. That is $0.75 = 7.5\times10^{-1}$. Its common logarithm is $\log 0.75 \approx -0.124$, showing it’s a small (negative) exponent because the value is less than 1.

Common mistake – Students often think the logarithm of a number < 1 should be positive because the number itself is “small”. Remember: the logarithm measures the power of 10 needed; a power less than 1 means a negative exponent.

Deeper note – This negative result is called the characteristic (the integer part) when the number is < 1; the fractional part is the mantissa.

In short: For any number smaller than 1, write it as $10^{\text{negative exponent}}$ and the common logarithm equals that negative exponent.