Logarithm: Exercises

All exercises should be done without a calculator. Sample solutions are below the section break.

Approximate the following logarithms. For each, find what integer it is smaller than, which it is larger than, and which of those it is closest to. How do you know? For extra challenge, find the first two digits (eg. 3.2) of each's decimal expansion.

1. ^[1]
2. ^[2]

If you'd like more logarithms practice, here's a practice sheet made by Claude Sonnet 4.6 based on the guide.

↓ Solutions ↓

Problem 1 Solution

Challenge:

 (real value ~1.893)

Problem 2 Solution

. 

I think you get the idea by now.

Challenge:

 (real value ~2.096)

Problem 3 Solution

Here we can use our solutions from the previous challenges:

 (real value ~3.989)

Problem 4 Solution

, hence .

Challenge:

To find the first two digits, notice that  (i.e. ), so:

(real value is ~3.636)

Problem 5 Solution

 (real value ~7.620)

1. ^{^}

   If you're going for the extra challenge, you'll need to know that 

2. ^{^}

   We'll make this one a bit harder: the value you'll need is .