# How do you evaluate log98.2?

Oct 11, 2017

See explanation

#### Explanation:

These days people use calculators. Years ago log tables were used and I am not sure if I can even find my old copy of one. If you wish to use log tables I did a quick search and found this site.

https://www.wikihow.com/Use-Logarithmic-Tables

The 9 in 98.2 is counting in tens so you will have a log value starting as $1.0$ plus some decimal that you put after the decimal point.

My calculator gives: 1.9921 rounded to 4 decimal places.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
What is $\log 98.2$ actually saying?

When you see it written like this it is generally accepted that you are using what is called base10. Really it should be written as ${\log}_{10} 98.2$

Suppose we set $\log 98.2 = x$

Then this is stating the condition that ${10}^{x} = 98.2$

EVALUATE means give value to. The value of log98.2 is

1.9921 rounded to 4 decimal places.

Oct 12, 2017

$\log \left(98.2\right) = \log \left(100\right) - \ln \frac{1 - 0.018}{\ln} \left(10\right) \approx 1.99211$

#### Explanation:

I will calculate this to just a few significant digits, but the same method can give you more using more terms...

Use:

$\ln 10 \approx 2.302585093$

$\ln \left(1 - t\right) = - t - {t}^{2} / 2 - {t}^{3} / 3 - {t}^{4} / 4 - \ldots$

So:

$\log \left(98.2\right) = \log \left(100 \cdot 0.982\right)$

$\textcolor{w h i t e}{\log \left(98.2\right)} = \log \left(100\right) + \log \left(1 - 0.018\right)$

$\textcolor{w h i t e}{\log \left(98.2\right)} = 2 + \ln \frac{1 - 0.018}{\ln} \left(10\right)$

Now:

$\ln \left(1 - 0.018\right) = - 0.018 - {0.018}^{2} / 2 - {0.018}^{3} / 3 - {0.018}^{4} / 4 - \ldots$

$\textcolor{w h i t e}{\ln \left(1 - 0.018\right)} \approx - 0.018 - \frac{0.000324}{2}$

$\textcolor{w h i t e}{\ln \left(1 - 0.018\right)} \approx - 0.018162$

So:

$\log \left(98.2\right) \approx 2 - \frac{0.018162}{2.3026} \approx 1.99211$

Oct 12, 2017

Use $\log \left(2\right) \approx 0.30103$ to find $\log \left(98.2\right) \approx 1.992$

#### Explanation:

Use:

$\log \left(2\right) \approx 0.30103$

Then:

$98.2 = 100 \cdot 0.982 = 100 \cdot \left(1 - 0.018\right) \approx \frac{100}{1 + 0.018} \approx \frac{100}{1.024} ^ \left(\frac{3}{4}\right) = \frac{100}{{\left({2}^{10} / {10}^{3}\right)}^{\frac{3}{4}}}$

So:

$\log \left(98.2\right) \approx \log \left(100\right) - \log \left({\left(\frac{{2}^{10}}{{10}^{3}}\right)}^{\frac{3}{4}}\right)$

$\textcolor{w h i t e}{\log \left(98.2\right)} \approx 2 - \frac{3}{4} \left(10 \log \left(2\right) - 3\right)$

$\textcolor{w h i t e}{\log \left(98.2\right)} \approx 2 - \frac{3}{4} \left(3.0103 - 3\right)$

$\textcolor{w h i t e}{\log \left(98.2\right)} \approx 2 - \frac{3}{4} \left(0.0103\right)$

$\textcolor{w h i t e}{\log \left(98.2\right)} \approx 1.992$