How do you use the Change of Base Formula and a calculator to evaluate the logarithm #log_2 15#?

Question

Ariana Alvarez · Accepted Answer

The Change of Base Formula merely states that: #log_color(green)(b) color(red)(a) = (log color(red)(a))/(log color(green)(b)) = (ln color(red)(a))/(ln color(green)(b))# (By the way, the equivalence to #ln a/ln b# just means that #lnx/logx# is a constant: #~~2.303#.) You just end up taking the base #10# or #e# logarithm of #a# and divide it by the same kind of #log# on #b#. The base changes so that both bases are the same. For example: Base 10 counting (decimal system)
#1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...# Base 2 counting (binary system)
#1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, ...# #1010# in base 2 (you can emphasize this as #1010_2#) is equal to #10# in base 10 (you can emphasize this as #10_10#).  To get two people to communicate well, they must speak the same language. To be able to divide two logarithms at all, you have to get the numbering systems to be the same. Conveniently, since #(log a)/(log b) = (ln a)/(ln b)#, you don't even need to specify the base of the "new #log#". All you do is: #log_2 15 = (log 15)/(log 2) = ln 15 / ln 2 = (ln (3*5))/ln 2 = color(blue)((ln 3 + ln 5) / ln 2 ~~ 3.907)#

Greyson Brown · Answer

undefined To evaluate the logarithm $\log_2 15$ using the Change of Base Formula and a calculator, follow these steps:

1. Use the Change of Base Formula, which states that $\log_b a = \frac{\log_c a}{\log_c b}$, where $a$ is the number you want to take the logarithm of, $b$ is the base of the logarithm you want to evaluate, and $c$ is any positive number not equal to 1.

2. Choose a base for the logarithm that your calculator supports. Most calculators have natural logarithm (base $e$, denoted as $\ln$) and common logarithm (base 10, denoted as $\log$) functions.

3. Rewrite $\log_2 15$ using the Change of Base Formula:

$$\log_2 15 = \frac{\log_{10} 15}{\log_{10} 2}$$

4. Use your calculator to find the values of $\log_{10} 15$ and $\log_{10} 2$.

5. Finally, divide the result of $\log_{10} 15$ by the result of $\log_{10} 2$ to obtain the value of $\log_2 15$.

By using the Change of Base Formula and a calculator, you can accurately evaluate the logarithm $\log_2 15$.