Q.The value of $\log_{10}1000$ is:

Expressing 1000 as a power of 10 reads the logarithm off directly as the exponent.

We need $\log_{10}1000$. Express 1000 as a power of the base 10:

Therefore, by the property $\log_{a}(a^{n}) = n$,

Why the other options are wrong:

• (b) 30 confuses the logarithm with something like $3\times10$ — a logarithm is the exponent (3), not the exponent times the base.
• (c) 100 is $10^{2}$, the number whose log is 2, not the log of 1000. …

Cross-check via the digit-count / characteristic idea. $1000$ is an exact power of 10, and $\log_{10}(10^{n}) = n$ for any integer $n$; since $1000 = 10^{3}$, the value is 3. Equivalently, …

The typical trap is choosing (d) $10^{3}$ because it "looks like" the answer — but $10^{3}$ is the number 1000, whereas its logarithm is the exponent 3. Confusing a number w …