Math  /  Algebra

Questionlog22.1\log _{2} 2.1

Studdy Solution

STEP 1

What is this asking? This is simply asking, "2 raised to *what power* gives us 2.1?" Watch out! Don't confuse logarithms with division or roots!
A logarithm is an exponent.

STEP 2

1. Rewrite the decimal
2. Approximate the logarithm

STEP 3

We're given log22.1\log_2 2.1.
Let's **rewrite** 2.1 as a fraction: 2.1=21102.1 = \frac{21}{10}.
So, we're looking for log22110\log_2 \frac{21}{10}.

STEP 4

Now, we can use the **logarithm property** that says log(ab)=log(a)log(b)\log(\frac{a}{b}) = \log(a) - \log(b).
This means we can rewrite our expression as log221log210\log_2 21 - \log_2 10.
This is super helpful because it breaks down our problem into smaller, more manageable pieces!

STEP 5

Let's think about log221\log_2 21.
We know that 24=162^4 = 16 and 25=322^5 = 32.
Since 21 is between 16 and 32, log221\log_2 21 must be between 4 and 5.
Let's **estimate** it to be around 4.4, since 21 is closer to 16 than to 32.

STEP 6

Now, let's tackle log210\log_2 10.
We know 23=82^3 = 8 and 24=162^4 = 16.
Since 10 is between 8 and 16, log210\log_2 10 must be between 3 and 4.
We can **estimate** it to be around 3.3, since 10 is a bit closer to 8 than to 16.

STEP 7

Remember, our expression is log221log210\log_2 21 - \log_2 10.
Using our **approximations**, we get 4.43.3=1.14.4 - 3.3 = 1.1.

STEP 8

So, log22.1\log_2 2.1 is approximately **1.1**.
This means 21.12^{1.1} is roughly equal to 2.1.

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