Step 15: Common Logarithms (3): At what decimal place does the first non-zero digit appear? Learning Logarithm Mechanisms in 1 Minute: 20-part series for mastering overwhelming knowledge
Hello, this is KoTa.
Imagine the number "(1/2) to the power of 100." If you multiply 0.5 by itself 100 times, you get 0.000... with a mind-boggling number of zeros.
You can also instantly find where this "non-zero digit" finally appears (at what decimal place) by using common logarithms.
The calculation method is almost the same as for finding the "number of digits."
Apply log_10 to the target number log_10 (1/2)^100
Bring the exponent to the front 100 × log_10 (1/2)
Substitute the value and calculate Since log_10 (1/2) is "-log_10 2," it is approximately -0.3010. 100 × (-0.3010) = -30.10
This negative number, "-30.10," is the answer. This value is greater than -31 and less than -30, isn't it?
According to the rules of logarithms, we take the absolute value of this smaller number (-31) to determine that the first non-zero digit appears at the **"31st decimal place."**
Huge numbers have positive logarithms, and tiny numbers have negative logarithms. By using logs, you can measure everything from the size of the universe to the size of a microorganism with the same ruler.
Key points for this lesson
The common logarithm of a number less than 1 (fractions or decimals) is a negative value.
If the calculation result is "-30.10," focus on the smaller integer, "-31."
The absolute value of that number is the answer to **"at what decimal place"**!
What is the next step?
Step 16: Logarithmic Functions (1): Features revealed by drawing graphs
Up until now, the focus has been on "calculation," but next, we will focus on "shape." What kind of curve does a log draw when plotted on a graph? Actually, it has a shape very similar to the "feeling" we get when we experience social media buzz or technological evolution.
See you then!